Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values 9789814689410, 9814689416

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Series on Number Theory and Its Applications*

ISSN 1793-3161

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Series Editor: Shigeru Kanemitsu (Kinki University, Japan) Editorial Board Members: V. N. Chubarikov (Moscow State University, Russian Federation) Christopher Deninger (Universität Münster, Germany) Chaohua Jia (Chinese Academy of Sciences, PR China) Jianya Liu (Shangdong University, PR China) H. Niederreiter (National University of Singapore, Singapore) Advisory Board: A. Schinzel (Polish Academy of Sciences, Poland) M. Waldschmidt (Université Pierre et Marie Curie, France)

Published Vol. 5 Algebraic Geometry and Its Applications edited by J. Chaumine, J. Hirschfeld & R. Rolland Vol. 6 Number Theory: Dreaming in Dreams edited by T. Aoki, S. Kanemitsu & J.-Y. Liu Vol. 7

Geometry and Analysis of Automorphic Forms of Several Variables Proceedings of the International Symposium in Honor of Takayuki Oda on the Occasion of His 60th Birthday edited by Yoshinori Hamahata, Takashi Ichikawa, Atsushi Murase & Takashi Sugano

Vol. 8 Number Theory: Arithmetic in Shangri-La Proceedings of the 6th China–Japan Seminar edited by S. Kanemitsu, H.-Z. Li & J.-Y. Liu Vol. 9 Neurons: A Mathematical Ignition by Masayoshi Hata Vol. 10 Contributions to the Theory of Zeta-Functions: The Modular Relation Supremacy by S. Kanemitsu & H. Tsukada Vol. 11 Number Theory: Plowing and Starring Through High Wave Forms Proceedings of the 7th China–Japan Seminar edited by Masanobu Kaneko, Shigeru Kanemitsu & Jianya Liu Vol. 12 Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values by Jianqiang Zhao *For the complete list of the published titles in this series, please visit: www.worldscientific.com/series/sntia

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Library of Congress Cataloging-in-Publication Data Names: Zhao, Jianqiang. Title: Multiple zeta functions, multiple polylogarithms, and their special values / by Jianqiang Zhao (Georgia Southern University, USA). Description: New Jersey : World Scientific, 2016. | Series: Series on number theory and its applications ; volume 12 | Includes bibliographical references and index. Identifiers: LCCN 2015043387 | ISBN 9789814689397 (hardcover : alk. paper) Subjects: LCSH: Functions, Zeta. | Logarithmic functions. Classification: LCC QA351 .Z43 2016 | DDC 515/.56--dc23 LC record available at http://lccn.loc.gov/2015043387

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谨以此书献给岚、盈、心雅 To Lan, Annie and Emma

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May 2, 2013 14:6 BC: 8831 - Probability and Statistical Theory

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Contents

xv

Preface

xvii

Introduction 1.

Multiple Zeta Functions 1.1 1.2 1.3 1.4 1.5

2.

1

Riemann Zeta Function . . Multiple Zeta Functions . . Analytic Continuation . . . Other Multiple Zeta Functions Historical Notes . . . . . Exercises . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

Multiple Polylogarithms (MPLs) 2.1 2.2 2.3

2.4 2.5 2.6

Chen’s Iterated Integrals . . . . . . . . . . Classical Polylogarithms . . . . . . . . . . Definition of MPLs . . . . . . . . . . . . 2.3.1 Divergence Control: DivLogN and DivLogD 2.3.2 Relations Between DivLogN and DivLogD . 2.3.3 Application of Iterated Integrals . . . . . . Analytic Continuation of Multiple Logarithms . . Monodromy of Multiple Logarithms . . . . . . Variation Matrix for MPLs . . . . . . . . . 2.6.1 General Procedure Leading to Pseudo Row Entries . . . . . . . . . . . . . . . . . . . . 2.6.2 Some Examples in Weight Four . . . . . . 2.6.3 A General Result . . . . . . . . . . . . . . vii

1 2 3 6 9 12 13

. . . . . . . . .

. . . . . . . . . . . .

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13 14 16 17 18 21 22 25 30

. . . . . . . . . . . .

31 32 36

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2.8*

2.9

2.10 2.11

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Variations of Mixed Hodge Structures of Multiple Polylogarithms . . . . . . . . . . . . . . . 2.7.1 Definition: A Review . . . . . . . . . . . . . 2.7.2 Mixed Hodge Structures of Multiple Logarithms . . . . . . . . . . . . . . . . . . . Limit Mixed Hodge Structures of Multiple Logarithms 2.8.1 Limit Mixed Hodge Structures of Double Polylogs . . . . . . . . . . . . . . . . . . . . . Single-Valued MPLs of Multi-Variables . . . . . . 2.9.1 General Procedure of Construction . . . . . . 2.9.2 Single-Valued Double Logarithms . . . . . . 2.9.3 Single-Valued Double Polylogs Li1,2 and Li2,1 . . . . . . . . . . . . . . . . . . . . . . Aomoto Polylogs . . . . . . . . . . . . . . Historical Notes . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . .

. . . . .

38 38

. . . . .

41 42

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. . . .

44 46 46 47

. . . . .

. . . .

48 50 55 58

Multiple Zeta Values (MZVs) 3.1 3.2

3.3

3.4 3.5 3.6

4.

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Multiple Zeta Functions, Multiple Polylogarithms, and Their Special Values

2.7

3.

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Shuffle Product and Euler’s Decomposition Formula Hoffman’s Algebraic Setup . . . . . . . . . 3.2.1 Shuffle Relations . . . . . . . . . . . . . . . 3.2.2 Stuffle Relations . . . . . . . . . . . . . . . Double Shuffle Relations . . . . . . . . . . 3.3.1 Lyndon Words . . . . . . . . . . . . . . . . 3.3.2 Regularization in Two Ways . . . . . . . . 3.3.3 Main Statement . . . . . . . . . . . . . . . Dimension Conjectures Over Q . . . . . . . . Linearized Double Shuffle and Parity Principle . . Historical Notes . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . .

63 . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . .

Drinfeld Associator and Single-Valued MZVs 4.1

Drinfeld Associator . . . . . . . . . . . . 4.1.1 KZ-Equations: A General Setup . . . . . . 4.1.2 Some Hopf Algebras and Their Group-Like Elements . . . . . . . . . . . . . . . . . . . 4.1.3 Grothendieck–Teichm¨ uller Group . . . . . 4.1.4 Pentagon and Hexagon Relations . . . . . .

63 64 64 68 68 69 74 79 80 83 89 93 97

. . . . . . .

97 97

. . . . 100 . . . . 104 . . . . 107

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4.2 4.3 4.4 4.5 4.6

5.

. . . .

112

. . . . .

116 119 124 128 129

. . . . .

. . . . .

. . . . .

Multiple Zeta Value Identities 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

5.10 5.11

6.

Single-Variable Multiple Polylogs . . . . . . Single-Valued Single-Variable Multiple Polylogs (SVMPLs) . . . . . . . . . . . . . . Single-Valued MZVs and Integration of SVMPLs Generating Series of Hoffman Elements . . . . Historical Notes . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . .

Duality Relations . . . . . . . . . . . . . . Cyclic Sum Relation . . . . . . . . . . . . . Euler’s Decomposition Formula Revisited . . . . . Weighted Sum Relations . . . . . . . . . . . Derivation Relations . . . . . . . . . . . . . Fixed Weight, Height and Depth Relations . . . . Zagier’s 2-3-2 Formula of MZVs . . . . . . . . An Exotic Shuffle Relation . . . . . . . . . . Period Polynomial Relations of Double Zeta Values . 5.9.1 Double Zeta Space . . . . . . . . . . . . . . . 5.9.2 Period Polynomials . . . . . . . . . . . . . . 5.9.3 Modular Forms and Cusp Forms on SL2 (Z) . 5.9.4 Even Weight Double Zeta Value Relations . 5.9.5 Odd Weight Double Zeta Value Relations . . Conjectural Multiple Zeta Value Relations . . . . . Historical Notes . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . .

135 . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

Symmetrized Multiple Zeta Values (SMZVs) 6.1 6.2 6.3 6.4*

6.5

Definition. . . . . . . . . . . . . . . . . Identities Involving SMZVs . . . . . . . . . . Drinfeld Associator and SMZVs . . . . . . . . Space Generated by SMZVs . . . . . . . . . . 6.4.1 Natural SMZVs . . . . . . . . . . . . . . . . 6.4.2 An Embedding ιn : Sn+1 ,→ GLn (Z) . . . . 6.4.3 Parity Results . . . . . . . . . . . . . . . . . 6.4.4 Diagonal Translation Invariant Polynomials . 6.4.5 SMZVs Generate Whole MZV Space . . . . . Historical Notes . . . . . . . . . . . . . .

135 136 140 141 152 159 162 166 170 170 171 174 177 183 188 189 195 199

. . . . . . . . . . . . . . .

. . . . . . . . . .

199 202 206 210 211 212 213 217 221 223

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Multiple Zeta Functions, Multiple Polylogarithms, and Their Special Values

Exercises . . . . . . . . . . . . . . . . . . .

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7.

Multiple Harmonic Sums (MHSs) and Alternating Version 7.1 7.2 7.3 7.4 7.5 7.6

8.

. . . . with . . . . . . . .

. . . . . . . . . . . . Signed Powers . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

225 . . . . . . .

. . . . . . .

Finite Multiple Zeta Values and Finite Euler Sums 8.1 8.2 8.3 8.4 8.5

8.6

8.7

9.

Definitions . . . . . . Stuffle Relations . . . . Quasi-Symmetric Functions Symmetric Sums . . . . Binomial Identities . . . Historical Notes . . . . Exercises . . . . . . .

Definitions . . . . . . . . . . . . . . . . . Reversal and Concatenation Relations . . . . . . . Stuffle Relations . . . . . . . . . . . . . . . Shuffle Relations . . . . . . . . . . . . . . . Finite Multiple Zeta Values (FMZVs) . . . . . . . 8.5.1 Homogeneous FMZVs . . . . . . . . . . . . . . . 8.5.2 Duality of FMZVs . . . . . . . . . . . . . . . . . 8.5.3 FMZVs of Lower Depths . . . . . . . . . . . . . 8.5.4 Structure of FMZVs . . . . . . . . . . . . . . . . 8.5.5 FMZVs of Superbity 2 . . . . . . . . . . . . . . . 8.5.6 FMZVs of Superbities ≤ 5 . . . . . . . . . . . . 8.5.7 FMZVs of Arbitrary Depths . . . . . . . . . . . Finite Euler Sums (FESs) . . . . . . . . . . . . 8.6.1 Two Reduction Formulas . . . . . . . . . . . . . 8.6.2 FESs of Small Depths . . . . . . . . . . . . . . . 8.6.3 FESs of Superbity 1 and 2 . . . . . . . . . . . . 8.6.4 Two Applications of MHS Binomial Identities . 8.6.5 FESs of Arbitrary Depths . . . . . . . . . . . . . Historical Notes . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . .

Definitions and Notation . . . Some q-Combinatorial Identities q-MH? S Identities . . . . . Some Applications . . . . .

. . . .

. . . .

. . . .

225 226 228 231 232 234 236 237

. . . . . . . . . . . . . . . . . . . .

q-Analogs of Multiple Harmonic (Star) Sums 9.1 9.2 9.3 9.4

223

238 241 244 245 247 247 250 253 254 255 260 261 262 262 263 268 269 271 272 275 279

. . . .

. . . .

. . . .

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279 281 284 290

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9.5 9.6

Congruences of q-MHSs . . . . . . . . . . . . . Historical Notes . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . .

10. Multiple Zeta Star Values (MZ?Vs) 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10

305

An Integral Expression . . . . . . . . Relation Between MZ? Vs and MZVs . . . Sum Relations . . . . . . . . . . . Cyclic Sum Relations . . . . . . . . Weighted Sum Relations . . . . . . . Fixed Weight, Height and Depth Relations Duality Relations . . . . . . . . . . Some Applications of Binomial Identities of Three-Two-One Formulas . . . . . . . Historical Notes . . . . . . . . . . Exercises . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . q-MZ? Vs . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

11. q-Analogs of Multiple Zeta Functions 11.1 11.2 11.3 11.4

Definitions . . . . . . . Analytic Continuation . . . Special Values at Non-Positive Historical Notes . . . . . Exercises . . . . . . . .

. . . . . . Integers . . . . . .

12.3 12.4 12.5 12.6 12.7

305 306 308 309 311 311 317 323 326 336 339 341

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

12. q-Analogs of Multiple Zeta (Star) Values 12.1 12.2

291 300 302

Various Definitions of q-MZVs . . . . Double Shuffle Relations . . . . . . 12.2.1 Rota–Baxter Algebras . . . . . 12.2.2 q-Analogs of Hoffman Algebras 12.2.3 q-Stuffle Relations . . . . . . . 12.2.4 Iterated Jackson q-Integrals . . 12.2.5 q-Shuffle Relations . . . . . . . Duality Relations . . . . . . . . . The P-R and R-P Relations. . . . . General Type G q-MZVs . . . . . . Application to Okounkov’s Conjecture . A q-Analog of Drinfeld Associator . . .

341 344 352 363 363 365

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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365 366 367 370 372 374 378 383 385 385 387 389

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12.8

q-MZVs by Bachmann and K¨ uhn . . . . . . . . . 12.8.1 Mono-Brackets . . . . . . . . . . . . . . . . . . . 12.8.2 A Derivation . . . . . . . . . . . . . . . . . . . . 12.8.3 Multiple Eisenstein Series . . . . . . . . . . . . . 12.8.4 Bi-Brackets and Double Shuffle Relations . . . . 12.9 Some Applications of MZ? V Binomial Identities . . . . 12.9.1 Mollified Companion of q-MZVs and q-MZ?Vs . 12.9.2 Applications to MZVs and MZ?Vs . . . . . . . . 12.10 Historical Notes . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

13. Colored Multiple Zeta Values

419

13.1 13.2 13.3

Definitions . . . . . . . . . . . . . . . . . Dimension Upper Bound by Deligne and Goncharov . . Standard Relations . . . . . . . . . . . . . . 13.3.1 Regularized Double Shuffle Relations . . . . . . 13.3.2 Proof of Regularized Double Shuffle Relations . 13.3.3 Weight One Relations . . . . . . . . . . . . . . . 13.3.4 Regularized Distribution Relations . . . . . . . . 13.3.5 Lifted Relations from Lower Weights . . . . . . 13.3.6 Group-Like Element I . . . . . . . . . . . . . . . 13.4* Generalized Drinfeld Associators . . . . . . . . . 13.5 Historical Notes . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . .

14. Colored Multiple Zeta Values at Lower Levels 14.1 14.2

14.3 14.4

EZ-Face: An Online Computation Resource . . . . Euler Sums . . . . . . . . . . . . . . . . 14.2.1 Dimension Conjectures . . . . . . . . . . . . 14.2.2 A Possible Integral Structure . . . . . . . . . 14.2.3 A Few Useful Integral Substitutions . . . . . 14.2.4 A Family of Identities . . . . . . . . . . . . 14.2.5 Generalized Doubling Relations at Level Two Nonstandard Octahedral Relations at Level Four . . Multiple Clausen and Deligne Values at Level Six . . 14.4.1 Definitions . . . . . . . . . . . . . . . . . . . 14.4.2 Central Binomial Sums . . . . . . . . . . . . 14.4.3 Dualities of Multiple Clausen Values . . . . .

390 390 395 400 401 405 406 408 410 412

419 420 421 421 425 434 435 438 440 441 444 446 449

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . .

449 451 451 453 455 455 460 461 464 464 467 469

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Contents

14.4.4 A 1-2-1 Formula of Multiple Glaishers . 14.4.5 Dimension Conjectures . . . . . . . . . 14.5* Level p and p2 Cases for Prime p ≥ 5. . . . . 14.6 Computational Results in Weight ≤ 5 . . . . 14.7 Historical Notes . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . .

. . . . . . . .

. . . . . .

. . . . . . . .

. . . . . .

15. Application to Feynman Integrals 15.1 15.2 15.3

15.4

15.5

15.6

470 474 476 481 483 485 489

Why Feynman Integrals? . . . . . . . . . . . Feynman Graphs . . . . . . . . . . . . . . Feynman Integrals in Momentum Space. . . . . . 15.3.1 Divergence Types . . . . . . . . . . . . . . . 15.3.2 A Short Example . . . . . . . . . . . . . . . 15.3.3 Massive and Massless Integrals . . . . . . . . Graphical Functions in Position Space . . . . . . 15.4.1 Sequential Graphs and Graphical Functions . 15.4.2 Properties of Graphical Functions . . . . . . Feynman Integrals in Parametric Form . . . . . . 15.5.1 Graph Polynomials in Edge Variables . . . . 15.5.2 Feynman Integrals in Projective Space . . . . Historical Notes . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

489 490 491 492 492 493 493 494 497 507 507 508 510 512

Appendix A

Key Concepts of Hopf Algebras

515

Appendix B

Some Useful Results from Lie Algebras

521

Appendix C

Basics of Hypergeometric Functions

529

C.1 C.2

Solutions of Some Ordinary Differential Equations . Transformation Formulas . . . . . . . . . . C.2.1 Linear Transformations . . . . . . . . . . . C.2.2 Some Transformations of 3 F2 . . . . . . . .

Appendix D D.1 D.2 D.3

. . . .

. . . . . .

. . . .

Sample Computer Codes

Double Shuffle Relations . . . . . . . . . . . . . Multiple Harmonic Sums . . . . . . . . . . . . . Finite Multiple Zeta Values . . . . . . . . . . . .

529 530 530 531 533 533 534 534

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Multiple Zeta Functions, Multiple Polylogarithms, and Their Special Values

Appendix E E.1 E.2

Multiple Zeta Functions, Multiple Polylo...

Tables of Special Values

Euler Sums of Lower Weights . . . . . . . . . . . Finite Multiple Zeta Values . . . . . . . . . . . .

Appendix F

Answers to Some Exercises

537 537 540 543

Bibliography

551

List of Abbreviations

585

List of Symbols

587

Index

591

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Preface

Since my postdoc years at UPenn, I have taken many notes from various papers to help myself keep abreast with the current research of the multiple zeta values and multiple polylogarithms. As the notes became thicker and thicker, I felt a strong urge to organize them into a book form so that not only my students and I but also other people interested in these objects would benefit from them. More recently, these objects have become more and more important due to their newly discovered close relations to numerous other branches of mathematics and even mathematical physics, so that such a textbook seems more appropriate. Originally I planned to include the most basic topics as well as some more advanced theories such as the p-adic multiple zeta values and the motivic fundamental groups. But as more details were written out, I found it very difficulty to maintain the book to an ideal length while keeping all of them together. Consequently, I must choose to stay on the more elementary side and insert only several optional sections (marked by *) for the more inquisitive reader. During the multi-year long writing process, I have benefited enormously from multiple visits to the Max Planck Institute for Mathematics, IHES, Taiwan National University, the Academia Sinica, and the Morningside Center of Mathematics in Beijing. This project was also supported partially by NSF grant DMS1162116. Many people have contributed to this work in various ways. Herbert Gangl was responsible for prompting me to prepare the first outline of this book in early 2010. He was also extremely helpful in providing with me many valuable opinions from the content of the book to the pictures on the cover. Thanks are also due to my students John Cutie and Margaret Sundberg, who suggested a lot of grammatical improvements. I want to express xv

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my heart-felt gratitude to Henrik Bachmann, Johannes Bl¨ umlein, David Bradley, David Broadhurst, Francis Brown, Tianxin Cai, Ching-Li Chai, Chieh-Yu Chang, Marc Conrad, Pierre Deligne, Kurusch Ebrahimi-Fard, Hidekazu Furusho, Herbert Gangl, Alexander Goncharov, David Goss, Li Guo, Richard Hain, Khodabakhsh and Tatiana Hessami Pilehrood, Hoang Ngoc Ming, Mike Hoffman, Masanobu Kaneko, Maxim Kontsevich, Ulf K¨ uhn, Peng Lei, W.-C. Winnie Li, Zhonghua Li, Erin Linebarger, Kohji Matsumoto, Dominique Manchon, Megan McCoy, Erik Panzer, Johannes Singer, Ismael Soud`eres, Roberto Tauraso, Danesh Thakur, Kevin Thielen, Jens Vollinga, Jos Vermaseren, Liuquan Wang, Fei Xu, Seidai Yasuda, Haiping Yuan, Don Zagier, Doron Zeilberger, Bin Zhang, Xia Zhou and Wadim Zudilin, for their valuable discussions, fruitful collaborations, and constant encouragement. I also want to thank Prof. Keqin Feng, Prof. Mike Rosen, and Prof. Jing Yu for their patience, insightful guidance, fatherly love, and immense help during the difficult times. Finally, I want to thank my wife for her unfailing love over the years, to my daughters for the joys they bring me, and to my parents for their sacrifice for me. This book would not be possible without their continuous support.

Jianqiang Zhao

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Introduction

The Riemann zeta function ζ(s) is one of the most fascinating objects in mathematics. It is a well-known historical fact that L. Euler had studied ζ(s) as a real function long before B. Riemann did. In particular, he was able to evaluate the function at positive even numbers up to 26 using Bernoulli numbers. He also established a few very nice relations among the double zeta values defined by the nested double sums X 1 , s, t ∈ N, ζ(s, t) := ms nt m>n≥1

which are said to have weight s + t. As a matter of fact, Euler worked with the double zeta star values ζ ? (s, t) which are defined by replacing m > n by m ≥ n in the above. Among his many amazing discoveries one stands out as the beautiful identity ζ(2, 1) = ζ(3), simple to state but not so easy to prove. If you have not seen it before, perhaps you should try it by yourself. Around 1990, more than 200 years after Euler, this subject was revitalized and generalized to the multiple zeta values (MZVs) by M. Hoffman and D. Zagier independently and almost simultaneously. These values provide the multiple variable version of zeta and double zeta values defined by the iterated sums X 1 ζ(s1 , . . . , sd ) := . (0.1) k1s1 · · · kdsd k1 >···>kd >0

It is important to note that another common way to define these in the literature is to use the increasing order on the indices. Since their debut MZVs have appeared unexpectedly in many diverse areas of mathematics and physics. Most prominently, they have shown up in the computation of some infinite classes of the Feynman integrals and knot invariants. Further, F. Brown proved in 2012 in [102] that all xvii

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the periods of the category of mixed Tate motives unramified over Z are 1 ]-linear combinations of the MZVs, revealing their fundamental nature Q[ 2πi in algebraic geometry. History often repeats itself. Just like Riemann considered ζ(s) as a complex function years after Euler’s work, multiple zeta functions ζ(s1 , . . . , sd ), where s1 , . . . , sd are complex variables, were investigated almost a decade after Hoffman and Zagier’s seminal works. We will visit this analytic theory via the Euler–MacLaurin summation formula in Chap. 1. In the mid 1990s, mathematical physicists discovered that one needs not only the MZVs to express some Feynman integrals but also the special values of the multiple polylogarithms (MPLs) at the sixth root of unity. Here, MPLs are multiple variable generalizations of the classical polylogarithms, which will be considered in Chap. 2. Besides the analytic properties, we also look at the mixed Hodge structures associated with MPLs and, as a by-product, sketch a way to define their single-valued versions. The next four chapters form the central part of this book. The reader will find many detailed results concerning the MZVs, proved using a variety of methods, such as the theory of hypergeometric series, Hopf algebras, and Lie algebras. Since these tools are rarely found in the standard graduate curriculum, we list a few key results and properties we need in the appendices found at the back of the book. One of the most fruitful ideas to study the rational structure of the MZVs is to consider so-called double shuffle relations. If two MZVs are multiplied, we can use either their infinite series representations or their iterated integral representations first observed by M. Kontsevich. The former leads to the quasi-shuffle structure while the latter yields the shuffle one. Comparing the two ways of computing the same product, one obtains so-called finite double shuffle relations. However, even in the simplest case, these relations are insufficient. For example, Euler’s identity ζ(2, 1) = ζ(3) cannot be proved this way simply because the smallest weight of a product of the MZVs is 4. To prove this identity, we are naturally led to the regularizations of the MZVs in two different ways corresponding to the two different product structures. We present this beautiful work of K. Ihara, M. Kaneko and D. Zagier [308] in Chap. 3. A surprising application of the MZV theory is that they can be used to express the Drinfeld associator which plays a key role in V. Drinfeld’s work on the deformation of some Hopf algebras related to the quantum Yang–Baxter equations [172, 173]. The first half of Chap. 4 centers around this key connection, which provides us with an invaluable framework to

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study the symmetrized MZVs in Chap. 6. The second half introduces a single variable cousin of MPLs, whose single-valued version will be crucial in evaluating several classes of the Feynman integrals in Chap. 15. Chapter 5 is a sort of compendium of various identities involving the MZVs whose proofs require many different methods. Chapter 6, on the contrary, is rather short and concentrates on only one special variant of MZVs constructed by a kind of cyclic symmetry. These symmetrized MZVs will be linked to the finite version of the MZVs to be considered in Chap. 8. When the infinite sums in Eq. (0.1) are truncated to finite partial sums, one obtains so-called multiple harmonic sums (MHSs), which will be treated in Chap. 7 together with their alternating version. As these sums are obviously rational numbers we are naturally led to examine their Van Hamme type congruence properties [292, 539, 619, 625]. Only recently do we realize that there should be close relations between the MZVs and their finite versions which are defined essentially by embedding the MHSs into a kind of ad`ele structure. In particular, this surprising connection motivates the study of the double shuffle relations among the finite MZVs and finite Euler sums. Note that a priori we have only one way to multiply the MHSs by their series definition and no integral expression is available. However, due to a beautiful conjecture of Kaneko and Zagier, we can now use some symmetrized versions of the MZVs to link to finite MZVs thereby hinting at a kind of double shuffle relations among them. This will be done in Chap. 8. For many interesting objects in number theory we often can and do consider their q-analogs. We will carry this out for the MHSs in Chap. 9. Many results contained in this chapter imply the corresponding ones for q-MZVs when we take the number of terms in the finite sums to infinity. When we further let q → 1− we arrive at statements for ordinary MZVs. And we can also extend this framework to their star version, which is the content of Chap. 10. In the next two chapters we continue to deal with q-analogs, first for the multiple zeta functions and then for their special values (q-MZVs). It transpires that there are quite a few different ways to define q-MZVs, which are often affiliated, although sometimes implicit at the first glance. Our attention here is mostly devoted to the double shuffle structures among these q-MZVs. We return to MPLs in Chaps. 13 and 14 where we shall consider the special values of these functions at the N th roots of unity. We call these numbers colored MZVs of level N . Unlike MZVs, the double shuffle relations only provide some of the Q-linear relations between colored MZVs.

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Chaps. 13 explains how to obtain more relations from algebraic and geometric considerations. Chaps. 14 gives more detailed structural results for lower level colored MZVs. In the last chapter we provide an attractive and dynamic link recently found between the MZVs/MPLs and the mathematical physics: the study of the Feynman integrals. This is one of the most active research areas in mathematical physics in the past 20 years. We consider two valuable techniques in these computations. The first is the graphical functions of O. Schnetz [505] which are used to evaluate the boxed ladder graphs, the wheel graphs, and the zig-zag graphs by applying the theory of single-valued MPLs and single-valued MZVs developed in Chap. 4. The second is to use the Schwinger parametric form of the Feynman integrals to transform these to projective integrals, which is particularly effective in interpreting results concerning Feynman integrals in terms of algebraic geometric objects such as motives and the cosmic Galois group originally conjectured by P. Cartier and currently being developed by F. Brown. Our primary goal in writing this book is to provide a comprehensive and accessible introduction to the theory of the MZVs and MPLs for advanced undergraduate students, graduate students and researchers who are interested in learning this beautiful subject. Due to the rapid development of these subjects in recent years as well as space limitation, I have refrained from wading into a few advanced topics such as the motivic theory [102,164], the p-adic theory [48,49,216,218,224,319], the transcendence theory [35,105,642], the relations between MPLs to cluster algebras [243,469], and the generalizations to elliptic MPLs [39, 57, 94, 95, 106, 200, 424], to name just a few. The omission does not mean these are negligible. On the contrary, to have a thorough understanding of the MZVs/MPLs these are often indispensable tools. For example, there is a close connection between the p-adic MZVs and the finite MZVs which provides a very promising approach to proving some of the most fundamental conjectures according to the recent works of D. Jarossay [318, 319] and S. Yasuda. The results contained in this book are due to so many people that it is impossible to attribute each individually. However, I have tried to compile them in the last section of each chapter as some historical notes. Further, some relevant research works are often stated without proofs in these sections. Hopefully, the interested readers can explore further and find their own next research project by themselves. To facilitate the study, I also provide a few exercise problems at the end of each chapter. Most of these problems are straightforward but some are

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probably difficult. The serious readers are encouraged to try at least some of them, especially those statements that are cited in the main body of the book. The subject of the MZVs/MPLs is often very computationally intensive and has been a hot bed for experimental mathematics for a couple of decades [32, 66]. We thus provide quite a few computer programming related problems scattered throughout the book and sketch a few crucial computer pseudo-codes in an appendix. Also included in the appendices are a few tables of useful numerical data and the answers to a few selected exercise problems. Many people have read the manuscript. Although a lot of inaccuracies have been caught and corrected during the editing process, there will likely be some more errors, for which I am solely responsible. If you notice any please send an email to me: [email protected].

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Chapter 1

Multiple Zeta Functions

The multiple zeta functions are nested generalizations of the Riemann zeta function. In this chapter we start by summarizing briefly the most important results of the Riemann zeta function. Then we bring in one of our major players in this book — the multiple zeta function — and outline its main properties and problems. Some results will be generalized to higher levels in Chap. 13, but often with more advanced and different proofs. So, we still provide complete proofs of these results in this chapter. 1.1 Riemann Zeta Function There are numerous accounts in the literature about the Riemann zeta function defined as follows: ∞ X 1 , ζ(s) := s n n=1

Re(s) > 1.

As motivations for further generalizations, we list some of its most important properties below. • Euler product formula. For Re(s) > 1, Y
−1 ζ(s) = 1 − p−s . p:prime

• The functional equation. The Riemann zeta function can be analytically continued to a meromorphic function on the whole complex s-plane, which is holomorphic everywhere except for a simple pole at s = 1 with residue 1. Moreover, it satisfies  πs  ζ(s) = 2s π s−1 sin Γ(1 − s) ζ(1 − s). (1.1) 2 1

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• Special values. For any positive even number 2n,

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1 B2n (2πi)2n , ζ(2n) = − · 2 (2n)!

(1.2)

where B2n are Bernoulli numbers defined by the generating function ∞ X t tn = Bn . t e − 1 n=0 n!

(1.3)

For negative integers −n < 0, one has ζ(−n) = −

Bn+1 . n+1

(1.4) 0

(x) • Generating function of the special values. Let ψ(x) := ΓΓ(x) be the digamma function defined for all x unequal to a negative integer. Then for all |x| < 1 X ψ(1 − x) = − ζ(k)xk−1 − γ, (1.5) k≥2

where γ ≈ 0.57721566 . . . is Euler’s constant. See, for example, [6, § 6.3]. Moreover, ψ satisfies the reflection, recurrence and duplication relations ψ(x) = ψ(1 − x) − π cot(πx) = ψ(x + 1) − 1 1 ψ(2x) = ψ(x) + ψ(x + 1/2) + log 2, 2 2 and it has an important integral expression Z 1 1 − tx dt, ψ(x + 1) = −γ + 0 1−t

1 , x

(1.6)

(1.7)

• Zeros, the critical line, and the Riemann hypothesis. From Eq. (1.4) we see that ζ(s) has zeros at negative even integers (called trivial zeros). In his landmark 1859 paper [483] Riemann conjectured that all non-trivial zeros of ζ(s) must lie on the critical line Re(s) = 1/2. This is the famous “Riemann hypothesis”. 1.2 Multiple Zeta Functions The multiple zeta functions are multiple variable generalizations of the Riemann zeta function.

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Definition 1.2.1. For fixed positive integer d and d-tuple of complex variables s = (s1 , . . . , sd ), the multiple zeta function is defined by X (1.8) ζ(s) := k1−s1 · · · kd−sd . k1 >···>kd >0

To guarantee convergence, s must satisfy Re(s1 + · · · + sj ) > j for all j = 1, . . . , d where Re(s) is the real part of a complex number s (see Exercise 1.4). The number d is called the depth, denoted by dp(s). When all the variables are positive integers we call |s| := s1 + · · · + sd the weight. Furthermore, the multiple zeta star function is defined by X ζ ? (s) := k1−s1 · · · kd−sd . (1.9) k1 ≥···≥kd ≥1

Remark 1.2.2. (i) In the literature the order 0 < k1 < · · · < kd is used sometimes, which gives rise to ζ(sd , . . . , s1 ) in our notation. (ii) It is not hard to see that ζ ? (s) can be expressed using the multiple zeta functions of different depths: X ζ ? (s1 , . . . , sd ) = ζ(s1 ◦ · · · ◦ sd ) (1.10) where ◦ is either “+” or “,”. So we will consider only the multiple zeta functions in the rest of this chapter. Similar to the Riemann zeta function one can study both the algebraic theory of the special values of the multiple zeta functions at positive integers, called the multiple zeta values and the analytic theory of the multiple zeta functions. In this chapter we will consider mainly the analytic side and postpone the treatment of the multiple zeta values to Chaps. 3–6. 1.3 Analytic Continuation Since the multiple zeta function of depth d in Eq. (1.8) is defined only for complex arguments with certain restrictions it is essential to continue it analytically to the whole d-dimensional complex space in order to study special values at arbitrary integer arguments. Recall that the Bernoulli polynomials Bk (x) are defined by the generating function ∞

X tk text = B (x) k et − 1 k! k=0

and the “periodic Bernoulli polynomials” are defined by ek (x) := Bk ({x}), B

x ≥ 1,

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where {x} is the fractional part of x. Then we have (see [571, Ch. IX, Misc. Ex. 12]) X e2πinx ek (x) = −k! . (1.11) B (2πin)k n∈Z\{0}

Recall that the Bernoulli numbers satisfy Bk = Bk (1) if k ≥ 2 while B0 = 1 and B1 = −1/2. Lemma 1.3.1. For every positive integer M ≥ 2 and x > 1, we have 4M ! e . BM (x) ≤ (2π)M Proof. It follows from the fact that ζ(M ) ≤ ζ(2) = π 2 /6 < 2 for M ≥ 2. Let’s recall the classical Euler–Maclaurin summation formula [571, 7.21]. Let f (x) be any (complex-valued) C ∞ function on [1, ∞) and let m and M be two positive integers with M even. Then we have Z m M m  X X Br  (r−1) 1 f (n) = f (m)−f (r−1) (1) f (x) dx+ (f (1)+f (m))+ 2 r! 1 r=2 n=1 Z m 1 eM (x)f (M ) (x) dx. B (1.12) − M! 1 Suppose Re(s) > 1, f (x) = 1/xs . Taking m = ∞ and m = k respectively in Eq. (1.12) and then taking the difference we get: ∞ k X 1 X X 1 1 = − ns n=1 ns n=1 ns

n>k

∞

Z k

=

Z ∞ M X Br (r−1) 1 1 eM (x)f (M ) (x) dx f (k) − f (k) − B 2 r! M ! k r=2 Z M X eM (x) (s)M ∞ B 1 Br (s)r−1 − s + − dx, s+r−1 2k r! k M ! k xs+M r=2

f (x) dx −

=

1 (s − 1)ks−1

where (s)M

Γ(s + M ) = = Γ(s)



1, s(s + 1) · · · (s + M − 1),

if M = 0; if M > 0,

(1.13)

is the Pochhammer symbol. Here we have used the fact that Bk = 0 if k is odd ≥ 3. Setting (s)0 = 1 and (s)−1 = 1/(s − 1), one has Z M X 1 X eM (x) Br (s)r−1 (s)M ∞ B = − dx. (1.14) s s+r−1 n r! k M ! k xs+M r=0 n>k

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Multiple Zeta Functions

Theorem 1.3.2. Let d be a positive integer. Define ) ( s =1, s + s = 2, 1, −2m ∀m ∈ Z , 1 1 2 ≥0 . (1.15) Sd = (s1 , . . . , sd ) ∈ Cd and s1 + · · · + sj ∈ Z≤j ∀j ≥ 3. (i) If d = 1 then we have M

ζ(s) =

1 1 X Br+1 + + (s)r s − 1 2 r=1 (r + 1)! Z ∞ (s)M +1 eM +1 (x)x−s−M −1 dx. B − (M + 1)! 1

(1.16)

(ii) If d ≥ 2 then for all s = (s1 , . . . , sd ) ∈ Cd \ Sd and positive integer M > d + |Re(s1 )| + · · · + |Re(sd )| we have ζ(s) =

M X Br r=0

r!

(s1 )r−1 · ζ(s1 + s2 + r − 1, s3 , . . . , sd )

(s1 )M − M!

X k2 >···>kd >0

1 s2 k2 · · · kdsd

Z

∞

k2

eM (x) B dx. xs1 +M

(1.17)

These provide the analytic continuation of the multiple zeta function ζ(s) to a meromorphic function on Cd with simple poles given by Sd . Proof. We prove the theorem by induction on the depth d. When d = 1, taking k = 1 and replacing M by M + 1 in Eq. (1.14) we get Eq. (1.16). Since we may choose M arbitrary large, this gives the analytic continuation of ζ(s) to the whole s-plane. Now we assume d ≥ 2. In Eq. (1.8) we replace s1 , k1 , k2 by s, n and k, respectively. Then Eq. (1.17) follows directly from Eq. (1.14) if we can show that the series in Eq. (1.17) converges. Lemma 1.3.1 implies Z ∞ e X BM (x) 1 dx s2 k2 · · · kdsd k2 xs1 +M k2 >···>kd >0

≤

4M ! (2π)M (M −1+Re(s1 ))

X

1

M −1+Re(s1 )+Re(s2 ) Re(s3 ) k3 k2 >···>kd >0 k2

Re(sd )

· · · kd

which, by induction, converges absolutely whenever M > d + |Re(s1 )| + · · · + |Re(sd )|. This completes the proof of the theorem by induction.

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1.4 Other Multiple Zeta Functions In the 2002 Arizona Winter School, Deligne designed a project for some participating graduate students to find a good candidate in the ´etale theory corresponding to the multiple zeta function defined by Eq. (1.8), called the p-adic multiple zeta functions. Subsequently Furusho successfully defined padic multiple zeta values by generalizing Coleman’s construction of p-adic polylogarithms and explained his results with the Tannakian formalism in [216–218]. Moreover, Chatzistamatiou [127] showed that these values are p-adic integers. Over global fields, one can generalize the multiple zeta functions to a kind of multiple Dedekind zeta functions over arbitrary number field F . Let OF be the ring of integers of F and N (I) = |OF : I| the norm for an integral ideal I in F . Let d be a positive integer and F1 , . . . , Fd be d number fields. R. Masri [416] defined the multiple Dedekind zeta function over F1 , . . . , Fd as (after slightly change of ordering of the indices) X ζd ({Fi }di=1 ; s1 , . . . , sd ) := N (n1 )−s1 · · · N (nd )−sd . n1 ∈OF1 ,...,nd ∈OFd N (n1 )>···>N (nd )

By Riemann–Roch he proves that there are positive numbers ε1 , . . . , εd such that the series ζd ({Fi }di=1 ; s1 , . . . , sd ) is absolutely convergent and analytic in the region Re(s1 + · · · + sj ) > j + ε1 + · · · + εj ,

∀j = 1, 2, . . . , d.

In [611] we consider the case where F1 = · · · = Fd = F and define ζF (s1 , . . . , sd ) = ζd ({F }di=1 ; s1 , . . . , sd ). In [300] Horozov approached this problem from a different point of view. In particular, his version of the multiple Dedekind zeta functions has analytic continuations and their special values can be expressed using iterated integrals. It would be very interesting to obtain some numerical data concerning the following generalization of Zagier’s conjecture concerning Dedekind zeta functions (cf. Conjecture 2.2.1). Problem 1.4.1. For large enough positive integers n1 , . . . , nd such that ζF (n1 , . . . , nd ) converges, is there an expression of ζF (n1 , . . . , nd ) in terms of a determinant with entries given by Lin1 ,...,nd evaluated at F -rational points up to some factors determined only by the number field F (such as the discriminant, the number of real and complex embeddings, etc.)? Here Lin1 ,...,nd is the single-valued version of the multiple polylogarithms to be defined in Sec. 2.9.

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In another direction, Akiyama and Ishikawa [8] defined the multiple Hurwitz zeta function by X ζ(s1 , . . . , sd ; θ1 , . . . , θd ) = (k1 + θ1 )−s1 · · · (kd + θd )−sd , (1.18) Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values Downloaded from www.worldscientific.com by HONG KONG UNIVERSITY OF SCIENCE AND TECHNOLOGY on 04/04/17. For personal use only.

k1 >···>kd >0

where 0 ≤ θj < 1 for j = 1, . . . , d and derive its analytic continuation by using a modified Euler–Maclaurin summation formula. Murty and Sinha obtained the analytic continuation in [432] by using the binomial series expansion and Hartog’s Theorem in complex analysis. All of the authors above considered also the cognate multiple L-series X χ1 (k1 ) . . . χd (kd ) , (1.19) L(s1 , . . . , sd ; χ1 , . . . , χd ) := k1s1 · · · kdsd k1 >···>kd ≥1

where χj ’s are Dirichlet characters. These values are essentially the colored multiple zeta values which we will study in Chap. 13. Kelliher and Masri [345] realized the analytic continuation of Eq. (1.18) with an extension of the generalized function approach first adopted in [605] and further determined some residues explicitly. M. Hoang Ngoc et al. [286] also studied these functions and used a certain shuffle product structure to produce an algorithm for computing the special values of the multiple polylogarithms (see Chap. 2 for definition). Generalizing the ideas of Wu, Matsumoto and et al., Zhao and Zhou (see [418, 423, 635]) defined the Mordell–Tornheim L-functions by ∞ ∞ X X χ1 (k1 ) . . . χd (kd )χd+1 (k1 + · · · + kd ) ··· LMT (s; χ) := . (1.20) k1s1 · · · kdsd (k1 + · · · + kd )sd+1 k1 =1

kd =1

They derived the analytic continuation by utilizing so-called colored Mordell–Tornheim functions. When all the characters in Eq. (1.20) are principal these functions are nothing but the traditional Mordell–Tornheim zeta functions ∞ ∞ X X 1 ζMT (s1 , . . . , sd+1 ) := ··· . (1.21) sd s1 k1 · · · kd (k1 + · · · + kd )sd+1 k1 =1

kd =1

Zhou et al. provided some signed q-analogs of Eq. (1.21) in [638], Okamoto studied a parametrized analogue in [450] while Tsumura and Okamoto considered its alternating analogues in [551] and [453], respectively. For another type of the multiple zeta functions, Matsumoto (see [421]) replaces the factor kj + θj in the denominator of Eq. (1.18) by linear polynomials of k1 , . . . , kj . More generally, while pursuing his Ph.D. Essouabri considered the multiple zeta functions of the form X P1 (k)−s1 · · · Pd (k)−sd k=(k1 ,...,kd )∈Nd

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where Pj (k)’s are polynomials of d variables. In his thesis [206, 207] Essouabri established the analytic continuation in the case d = 1 and stated that his method could be applied to arbitrary depth for suitable Pj ’s including all the linear polynomials. Later, Essouabri et al. [208] extended this to more general types of the multiple zeta functions involving vectorial functions in the denominators. One of the most interesting multiple zeta functions originates from physics. Around 1990, related to his work on topological quantum field theory, Witten [573] found the volumes of the moduli spaces of representations of the fundamental groups of two dimensional surfaces are some special values of a new type of zeta functions attached to complex semisimple Lie algebras g at positive integers. Inspired by this result, Zagier [598] defined these complex functions as follows: X 1 , ζW (s; g) := (dim ρ)s ρ where ρ runs over all finite dimensional irreducible representations of g. By physical considerations Witten showed that for any positive integer n ζW (2n; g) = c(2n; g)π 2nr , where c(2n; g) ∈ Q and r is the number of positive roots of g. Such formulas are now called Witten volume formulas. Many results along this direction can be found in [621, 624, 626, 636]. Matsumoto and his collaborators [354–357, 422] generalized Witten’s zeta functions to the multiple variable setting over an arbitrary semi-simple Lie algebra g. Let ∆+ be the set of positive roots of g, {α1 , . . . , αd } the fundamental roots with coroots {α1∨ , . . . , αd∨ }, and {λ1 , . . . , λd } the fundamental weights such that hαi∨ , λj i = δi,j (the Kronecker symbol). Then the multiple Witten zeta function over g is defined by ∞ X Y ζg ({sα }α∈∆+ ) := hα∨ , k1 λ1 + · · · + kd λd i−sα , (1.22) k1 ,...,kd =1 α∈∆+

which satisfies (see [354]) ζW (s; g) = ζg ({s}d )

Y α∈∆+

d

hα∨ , λ1 + · · · + λd i,

where {s} means the variable s repeats itself d times. One of the most exciting results is a statement on their functional relations produced by the Weyl group symmetry in the underlying Lie algebra structure (see [352,357] for details). For other relevant works, see [451, 452].

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It is well known that there is a huge supply of analogies between the number fields and the function fields over finite fields. Carlitz and Weil worked on these in the early years of twentieth century, continued by many researchers recently (see [255, 545] for a lot of fascinating results along this line). In particular, Thakur devoted a section of his book [545, § 5.10] to the study of various possible multiple zeta values over the rational function field Fq [t] over the finite field Fq of q elements. For global function field K over Fq , we now state a result of Masri [417]. For a divisor D of K, let + deg(D) be its degree and |D| = q deg(D) be its norm. Let DK be the sub semi-group of effective divisors of K. Masri defines d Y X |Dk |−sk , Zd (K; s1 , . . . , sd ) := + k=1 + ×···×DK (D1 ,...,Dd )∈ DK 0≤deg(D1 )≤···≤deg(Dd )

and shows that it has a meromorphic continuation to all of Cd and is a rational function in each of q −s1 , . . . , q −sd with a specified denominator. Further, Zd (K; s1 , . . . , sd ) has all possible simple poles on the linear subvarieties sk + · · · + sd = 0, 1, . . . , d − k + 1,

k = 1, . . . , d.

It has been discovered that Thakur’s multiple zeta value over the function fields enjoy some similar identities to those over Q (see [11, 125, 129, 130, 379–382]). Due to the lack of integral representation of the function field multiple zeta values, one loses the important family of so-called double shuffle relations (see Chap. 5). However, on the transcendence of these values Yu’s classical results in [588] and the new breakthrough by Chang [124] goes far beyond what we can do over the number fields. 1.5 Historical Notes Despite its name the Riemann zeta function was studied first by Euler before the mid eighteenth century (see, for example, [211]) although he considered it as a function of a real variable. It is Riemann who treated it as a complex function for the first time in his famous 1859 article [483]. The genesis of the multiple zeta functions can be traced back to a series of correspondences between Leonhard Euler and Christian Goldbach (see [323]). On the Christmas Eve of 1742, with different notation Goldbach wrote down some special cases of the following infinite sum on a letter to Euler: X 1 , am bn a≥b≥1

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where m and n are positive integers. In modern notation this is ζ(m, n) + ζ(m+n) where ζ(m, n) is called a double zeta value. Later, Euler discovered so-called decomposition formula for the double zeta values: 2ζ(n, 1) = nζ(n + 1) −

n−2 X i=1

ζ(n − i)ζ(i + 1),

(1.23)

for all positive integer n ≥ 2. There are two independent sources of the revival of the multiple zeta functions in recent years. From late 1980s to early 1990s, motivated by some problems of Moen (see [297]) Hoffman initiated the study of the multiple zeta values from the viewpoint of their underlying algebraic structures in a series of papers (see [288, 289]). During the same time period, Zagier was attracted to the multiple zeta functions by their mysterious connections to Witten’s zeta functions, Vassiliev knot invariants and the theory of mixed Tate motives over Z. After many discussions with Drinfeld, Kontsevich and Goncharov, Zagier included section 9 in his 1994 paper [598] which became one of the most influential papers in the modern study of the multiple zeta functions. However, the analytic properties of the double zeta function were considered much earlier than the general multiple variable case, similar to the situation of Euler’s study of the double zeta values at positive integers. The analytic continuation of the double zeta function was first realized by Atkinson [23] using the Poisson summation formula in the 1940s, and by Apostol and Vu in [20] independently much later. For arbitrary depth, there are many different approaches: the generalized functions or distributions of [462,605], the Euler–Maclaurin summation formula of [10,160], the Mellin–Barnes integral [419, 420], the multiple Hurwitz–Lerch zeta functions [351], and the binomial series expansion and Hartog’s Theorem [432]. In Sec. 1.3 we gave a polished proof of the second approach using the Euler– Maclaurin summation formula outlined first in [615]. Most of the current research on multiple zeta functions is on their special values. However, there are some important results concerning their analytic properties. For example, Matsumoto showed a functional equation for the double zeta functions (see [142, 420]) and Nakamura and Pankowski discovered the existence of nontrivial zeros of general multiple zeta functions in [435, 437] although their precise locations are still beyond our reach. For a good survey of the analytic theory of these functions and further generalizations, Matsumoto’s paper [418] is a good reference for results up to around 2001.

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Multiple Zeta Functions

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There are some investigations of the behavior of the multiple zeta functions at non-positive integers. At depth greater than 1, strictly speaking, different ways to define these values may lead to different results. This phenomenon already occurs at depth two, see Exercise 1.6. In [9, 10], Akiyama et al. provided several possible definitions. Komori [351] and Sasaki [497, 498] also considered this problem from different perspectives. By Connes–Kreimer’s algebraic approach with Rota–Baxter algebras, Guo and Zhang [264] first defined renormalized MZVs at non-positive integers satisfying stuffle relations. Manchon and Paycha [414] further provided a way to renormalize all MZVs at singular points. By choosing appropriate finite “linear” combination of multiple zeta-functions with some argument shifts, Furusho et al. [225] managed to desingularize the multiple zeta functions, which yields a rigorous definition of multiple zeta functions at non-positive integers. In [184], we showed that all the different ways to renormalize MZVs satisfying the stuffle relations form a set on which a certain transfer group acts freely and transitively. Using Raabe’s identity, Sadaoui [490] linked a certain multiple integral and a classical inversion argument to obtain an analytic continuation of the multiple zeta function defined at negative integer arguments. This process has been used to derive some closed expressions for these values by Moll et al. [427]. In the literature there exist a few functions named as “multiple zeta functions” but are defined completely differently from Eq. (1.8). For example, Kurokawa’s multiple zeta function considered in [367] is a tensor product of the general zeta functions that are defined by Euler products and are meromorphic of finite order with functional equations over the entire space. In particular, the tensor product of two zeta functions is defined by the Hadamard product with zeros and poles given by the sum of those of the two zeta functions. So Kurokawa’s multiple zeta function differs drastically from ours since we do not have Euler product expansion associated with our version (see [436] for a proof and [491] for a related work on multiple zeta-star functions). In [473] one can also find the multiple zeta functions of the free Abelian groups. A caveat in reading papers on multiple zeta functions is that sometimes the word “multiple” is used to refer to multiple summation indices even though the function is a single variable function. For instances, the p-adic multiple zeta function of [535] and the Barnes multiple zeta-function of [353] are both single variable functions.

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Exercises

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1.1. Show that B2n+1 = 0 for all n > 0 by the definition Eq. (1.3). 1.2. Prove Eq. (1.4) using the functional equation given by Eq. (1.1) and Eq. (1.2). 1.3. Prove the relation between multiple zeta star function and the multiple zeta function given by Eq. (1.10). 1.4. Show that Eq. (1.8) converges if and only if Re(s1 + · · · + sj ) > j for all j = 1, . . . , d. 1.5. Complete the proof of Thm. 1.3.2 by induction. 1.6. Carry out the analytic continuation of the double zeta function ζ(s1 , s2 ) explicitly. Then compare the two different limits lim lim ζ(s1 , s2 )

s1 →0 s2 →0

and

lim lim ζ(s1 , s2 )

s2 →0 s1 →0

at (s1 , s2 ) = (0, 0). Are they the same? Why?

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Chapter 2

Multiple Polylogarithms (MPLs)

In this chapter we consider single- and multi-variable generalizations of the natural logarithm, called the polylogarithms (polylogs) and the multiple polylogarithms (multiple polylogs or MPLs), respectively. 2.1 Chen’s Iterated Integrals We briefly recall the theory of K.-T. Chen’s iterated integrals in this section. It is an indispensable tool in the subsequent study of the analytic properties of MPLs. For r > 1, define inductively Z τ  Z b Z b f1 (t) dt · · · fr (t) dt = f1 (τ ) dτ · · · fr−1 (τ ) fr (t) dt dτ. a

a

a

When r = 0 one sets the integral to be 1 as a convention. More generally, let w1 , . . . , wr be some 1-forms (repetition allowed) on a manifold M and let α : [0, 1] → M be a piecewise smooth path. Write α∗ wi = fi (t)dt and define the iterated integral Z Z 1 w1 · · · wr := f1 (t) dt · · · fr (t) dt. α

0

Remark 2.1.1. Our order of the 1-forms in the iteration here is opposite to K.-T. Chen’s original order. The following results are crucial in the application of Chen’s theory of the iterated path integrals. However, the proofs are beyond the scope of this book so we recommend the interested reader to read K.-T. Chen’s original papers. Lemma 2.1.2. Let wi (i ≥ 1) be C-valued 1-forms on a manifold M . Then R (i) The value of α w1 · · · wr is independent of the parametrization of α. 13

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(ii) If α, β : [0, 1] −→ M are composable paths ( i.e., α(1) = β(0)), then Z Z r Z X w1 · · · wr = w1 · · · wj wj+1 · · · wr , Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 06/12/16. For personal use only.

βα

j=0

β

α

R

where we set α φ1 · · · φm = 1 if m = 0 and βα denotes the composition of α and β. (iii) For every path α : [0, 1] −→ M , let α−1 (t) = α(1 − t). Then Z Z r w1 · · · wr = (−1) wr · · · w1 . α−1

α

(iv) Let Sn be the symmetry group of n letters. For every path α, Z α

w1 · · · wr

Z α

wr+1 · · · wr+s =

X

Z

α σ∈Sr+s σ(1)kd ≥1

xk11 xk22 · · · xkdd . k1s1 k2s2 · · · kdsd

(2.7)

As in the definition of the multiple zeta functions, the order of indices in Eq. (2.7) is sometimes reversed in research papers, even by the author himself. Note also that occasionally the term “multiple polylog” means the

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Multiple Polylogarithms (MPLs)

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17

single variable function Lis1 ,s2 ...,sd (t, 1, 1, . . . , 1) in the literature, which will in fact be needed in this book later (see Sec. 4.2 and Sec. 14.1). Qi Now by setting ai = j=1 x−1 it is not hard to check (see Exercise 2.4) j that  sd −1 Z 1  s1 −1 dt dt dt dt Lis1 ,...,sd (x1 , . . . , xd ) = ··· . (2.8) t a − t t a −t 1 d 0 This is an iterated path integral in the sense of Chen (see [132, 133]) whose path lies in C. Thus one can easily enlarge its domain of definition to some open subset of Cd . However, it is difficult to study the monodromy of the MPLs by this integral expression. 2.3.1

Divergence Control: DivLogN and DivLogD

Let D = {z ∈ C : |z| < 1} be the open unit disk and H(D) be the set of holomorphic functions on D. Definition 2.3.2. Let CN be the set of infinite sequences in C. Let DivLogN ⊂ CN such that S = (S1 , S2 , . . . ) ∈ DivLogN if and only if there is an asymptotic expansion Sn = AsS (log n) + O(logα (n)/n)

as n → ∞,

where AsS (t) ∈ C[t] and α ∈ N. Example 2.3.3. In applications we often think S as coming from some partial sum sequence of an infinite series. For instance, it is well known that the partial sums of harmonic series, namely, the harmonic numbers Hn = 1 + 21 + · · · + n1 have the asymptotic formula Hn = log n + γ + O(log(n)/n)

as n → ∞,

where γ ≈ 0.57721566 . . . is Euler’s constant. Hence AsH (t) = t + γ for H = (H1 , H2 , . . . ). Similarly, for any n ∈ N0 , we have Asa(n) (t) = tn where a(n) = ((H1 − γ)n , (H2 − γ)n , . . . ). Thus As : DivLogN → C[t] is surjective. Definition 2.3.4. Let DivLogD be the set of functions f ∈ H(D) whose asymptotic expansion, as z → 1 in D ∩ R, is given by   f (z) = Asf (− log(1 − z)) + O (1 − z) logα (1 − z) , where Asf (t) ∈ C[t] and α ∈ N0 .

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Example 2.3.5. For any n ∈ N0 , if we take f (z) = (Li1 (z))n = (− log(1 − z))n then we see that AsLin1 (t) = tn . Thus As : DivLogD → C[t] is surjective. Remark 2.3.6. Define a ring structure on CN by the componentwise addition and multiplication. Then it is not hard to see that DivLogN becomes a sub-ring of CN . It is also clear that DivLogD is a sub-ring of H(D). 2.3.2

Relations Between DivLogN and DivLogD

For any sequence S = (S1 , S2 , . . . ), we define the formal power series associated with it as follows: X Fp(S) := un z n (2.9) n≥1

where Sn = u1 + · · · + un . Lemma 2.3.7. If S ∈ DivLogN and AsS = 0 then the associated power series f = Fp(S) ∈ DivLogD and Asf = 0. Proof. By the definition, there is α ∈ N such that Sn = u1 + · · · + un = O(logα (n)/n) as n → ∞. By Abel’s transformation we see that M X X X X un = (1−z) z M O(logα (M )/M ). f (z) = un z n = (z M −z M +1 ) n≥1

n=1

M ≥1

M ≥1

We claim that X M ≥1


z M O(logα (M )/M ) = O | logα+1 (1 − z)| .

(2.10)

To establish this, we break the infinite sum into two parts, under the assumption that z < 1 but is very close to 1. First, by the Integral Test Z 1/(1−z) X logα (t) (− log(1 − z))α+1 logα (M ) ≤ dt = . zM M t α+1 1 0 0 for sufficiently large m as logk (t)/t is decreasing when t > ek . Thus the series in Eq. (2.11) converges absolutely since |z| ≤ 1 and P um converges (see Exercise 2.2). If (s, z) = (1, 1), then the big-O term contributes to O(logα (n)/n2 ) which can be handled as above. So we only need to consider the leading

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term logk (n)/n. Using the decomposition log(n + 1) = log(n) + 1/n + O(1/n2 ) we see that logk+1 (n + 1) − logk+1 (n) logk (n) = + O(logk /n2 ). n k+1 Summing up we thus get the major term for TM as logk+1 (M )/(k + 1). Hence AsT (t) = tk+1 /(k + 1) whose derivative is tk . This finishes the proof of the proposition. We have a quantitative version of the case s = 1, z 6= 1 of Prop. 2.3.8. Lemma 2.3.9. For all k ≥ 0, we have    ∞ X logk m m−1 1 k+1 z = O log as z → 1 in D ∩ R. m 1−z m=1 Proof. We proceed with   induction   on k. If k = 0 then the left-hand side 1 1 is z −1 log 1−z = O log 1−z as z → 1 in D ∩ R. Assume the lemma holds for k = l ≥ 1 for some l. Then we have logl+1 m = (l + 1)

Z 1

m

m X logl x logl n dx ≤ 2(l + 1) . x n n=1

(2.12)

Here we have used the fact that the function f (x) = logl x/x is increasing on (1, el ) and then decreasing on (el , ∞). Hence if m < el then Eq. (2.12) following from a simple application of integral test technique (and the 2 is unnecessary). If m > el then suppose a < el < a + 1 for some integer a < m. We see that the area below f (x), 1 ≤ x ≤ m, can be covered by the following three kinds of rectangles: (1) a−1 rectangles, each based on [i−1, i] with height f (i), i = 2, · · · , a; (2) m − a − 1 rectangles, each based on [i, i + 1] with height f (i), i = a + 1, . . . , m − 1; (3) one rectangle based on [a, a + 1] with height f (el ). Pm logl n give the first m − 2 Note that the m − 1 nonzero terms in n=1 n l

rectangular areas in (1) and (2) and the last term logmm is not used. The reason is the value l/el is larger than all the values of the f (x) at the division points x = 2, · · · , m. So we need to add the one rectangle in (3) by doubling the sum. In fact we need only the two terms with n = 2 and 3

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Multiple Polylogarithms (MPLs)

since f (el ) = l/el < logl 2/2 + logl 3/3 for all l ≥ 1. From Eq. (2.12) we get

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∞ m ∞ X X logl+1 m m−1 z m−1 X logl n z ≤ 2(l + 1) m m n=1 n m=1 m=1

∞ ∞ X logl n n−1 X z r−1 z n r+n−1 r=1 n=1    X ∞ 1 1 logl n n−1 < 2(l + 1) log z . t 1−z n n=1

= 2(l + 1)

Thus the lemma follows from the induction assumption. For M ∈ N, we may define the M th partial sum of a MPL as X

) Li(M s1 ,...,sd (z1 , . . . , zd ) :=

M ≥k1 >···>kd >0

z1k1 · · · zdkd . k1s1 · · · kdsd

(2.13) d

Corollary 2.3.10. For all s = (s1
, . . . , sd ) ∈ Nd and z = (z1 , . . . , zd ) ∈ D , ) (z) M ≥1 ∈ DivLogN . It converges if and only the sequence Li(s; z) := Li(M s if (s1 , z1 ) 6= (1, 1). Proof. This follows from Prop. 2.3.8 and an induction on the depth d. 2.3.3

Application of Iterated Integrals

Let a1 , . . . , ak ∈ C and fix a path γ : [0, 1] → C. Define Z Iγ (a1 , . . . , ak ) := 0

1

Z

∗

t1

(γ ωa1 )(t1 ) (γ ∗ ωa2 )(t2 ) · · · 0 Z tk−2 Z tk−1 ∗ (γ ωak−1 )(tk−1 ) (γ ∗ ωak )(tk ), (2.14) 0

0

where, for all z ∈ C dt , −1 − t z ωz (t) = dt   , t   

if z 6= 0; if z = 0.

−1 By Lemma 2.1.3 we see that if the image of γ ∈ C \ {a−1 1 , . . . , ad } then Iγ is well defined and depends only on the homotopy class of γ. By convention, we set Iγ = 0 if k = 0. Also, for any (α, β) ∈ C2 , we denote by [α, β] the straight path t 7→ α + (β − α)t, t ∈ [0, 1].

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Proposition 2.3.11. Let d ∈ N, s = (s1 , . . . , sd ) ∈ Nd and z1 , . . . , zd ∈ Qk D \ {0}. Set zk0 = j=1 zj for k = 1, . . . , d. Then for all z ∈ D, Lis (zz1 , z2 , . . . , zd ) = I[0,z] ({0}s1 −1 , z10 , {0}s2 −1 , z20 , . . . , {0}sd −1 , zd0 ). Proof. This is essentially Eq. (2.8). See Exercise 2.4. 2.4 Analytic Continuation of Multiple Logarithms In this section we describe a method of defining the analytic continuation of multiple logarithms by Chen’s iterated integrals. This can be generalized to arbitrary MPLs but the notation is too cumbersome so we leave it to the interested reader (see [612]). For any positive integer n, we set Ln (x) = Li 1,...,1 (x) and define the |{z} n times

index set In := {0, 1}n = {i = (i1 , . . . , in ) : it = 0 or 1 ∀t = 1, . . . , n}.

(2.15)

Put 0 = (0, · · · , 0) and 1 = (1, . . . , 1). The weight function on the indices Pn |(i1 , . . . , in )| = t=1 it is the number of nonzero components. Lemma 2.4.1. Let x = (x1 , . . . , xn ) be an ordered set of complex variables. Let xj = (x1 , . . . , xj−1 , xj xj+1 , xj+2 , . . . , xn ) for j = 1, . . . , n − 1 and xn = (x1 , . . . , xn−1 ). Then the total differential dLn (x) = =

n X

dj Ln (x) j=1 n  X Ln−1 (xj ) j=1

=

n X j=1

1 − xj

Ln−1 (xj−1 ) dxj xj (xj − 1) ! 1 − x−1 j+1 , 1 − xj

dxj +

Ln−1 (xj )d log

 (2.16)

(2.17)

where xn+1 = ∞, L0 (x1 ) = 1 and when t = 1 the second term in the sum of Eq. (2.16) does not appear.

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Multiple Polylogarithms (MPLs)

23

Proof. Suppose j > 1 (the case j = 1 is simpler). Then by the definition c k k X X−1 k −1 xk11 · · · xj j · · · xknn j−1 dj Ln (x) = xj j dxj b k1 · · · kj · · · kn kj =kj+1 +1 c k1 >···>kj >···>kn

X

=

c k1 >···>k j >···>kn

c k k k −1 xk11 · · · xj j · · · xknn xj j−1 − xj j+1 · dxj xj − 1 k1 · · · kbj · · · kn

Ln−1 (xj−1 ) Ln−1 (xj ) dxj , dxj + 1 − xj xj (xj − 1) where we have set kn+1 = 0. This proves Eq. (2.16). Equation (2.17) now follows immediately (see Exercise 2.5). =

Suppose i = (i1 , . . . , in ) has weight k and let 1 ≤ τ1 < · · · < τk ≤ n such that iτ1 = · · · = iτk = 1. We define τm Y x(i) = y = (y1 , . . . , yk ), where ym = xα , 1 ≤ m ≤ k (2.18) α=1+τm−1

with τ0 = 0. Set w0 (x) = 0 and  1 − x−1  t+1 wt (x) := d log , for 1 ≤ t ≤ n. 1 − xt We now define a partial order ≺ on In as follows: Given i = (i1 , . . . , in ) and j = (j1 , . . . , jn ), j ≺ i (or, equivalently, i  j) if jt ≤ it for every 1 ≤ t ≤ n and js < is for some s. Theorem 2.4.2. Let x = (x1 , . . . , xn ) be an ordered set of complex variables. The multiple logarithm Ln (x) is a multi-valued holomorphic function on     Y Y
Sn0 = Cn \ (x1 , . . . , xn ) : (1 − xj ) 1 − xj · · · xk = 0   1≤j≤n

1≤j I1 (wj ) = I1 (wj+1 ) = · · · = I1 (w` ) > I1 (w`+1 ) then we compare Ij (wα ) and Ij (wα+1 ) for all α = 1, . . . , ` − 1 and for increasing j = 2, 3, · · · , until we find either of the two: (i) Ij−1 (wα ) = Ij−1 (wα+1 ) 6= 0 but Ij (wα ) 6= Ij (wα+1 ). If Ij (wα ) > Ij (wα+1 ) > 0 or Ij (wα ) = 0 < Ij (wα+1 ) then we concatenate wα and wα+1 , otherwise we don’t. Here Ij (wα ) = 0 means wα has depth j − 1 only. (ii) wα = wα+1 . Then we don’t concatenate them. Now it is not hard to see we do have non-increasing Lyndon words in the factorization. Example 3.3.6. We have the following Lyndon factorizations (x1 )(x0 ), (x1 )(x0 x1 )(x30 x1 x30 x1 x0 x1 )(x0 ), (x30 x1 x0 x1 )(x30 x1 x20 x1 ),

(x20 x1 )(x30 x1 x20 x1 x30 x1 x0 x1 ),

(x30 x1 x0 x1 )(x30 x1 ).

Our next goal is to show that every word w ∈ A1∗ = (A1 , ∗) has a unique expression as a Q-linear combination of the stuffle products of nonincreasing Lyndon words. Lemma 3.3.7. For 1 6= w ∈ A, let LW(w) be the largest word appearing in w. Let w, u, v ∈ A such that u < v. Then LW(u ∗ w) < LW(v ∗ w). Proof. Use induction on the weight sum |u| + |v|. Lemma 3.3.8. Let u, v ∈ A such that u is a Lyndon word and u ≥ v. Then LW(u ∗ v) = uv. Proof. Since x0 commutes with every word under stuffle we may assume that u = zs1 · · · zsd and v = zt1 · · · zt` ∈ A1 without loss of generality. Recall that in u ∗ v there are two kinds of words, one is obtained by pure shuffle, while the other involves stuffing. Assume s1 = t1 , . . . , si = ti and, 0 < si+1 < ti+1 or ti+1 = 0 since u ≥ v. Then in each term of the shuffle,

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71

except for uv, zt1 appears between zsj−1 and zsj for some 1 ≤ j ≤ d (zs0 is vacuous). If j > i then clearly sj < s1 ≤ t1 since u is Lyndon word and thus this term is smaller than uv. If j ≤ i then all of zsj up to zsi in this shuffle term are pushed to the right and thus producing a smaller term than uv since uv’s zsi+1 -term corresponds to some ztk or some zsk with k ≤ i. Finally, if some stuffing is involved, then t1 , . . . , ti cannot become smaller resulting a stuffing term smaller than uv. This shows that LW(u ∗ v) = uv and the lemma is proved. Lemma 3.3.9. Let u, v ∈ A1 such that u is a Lyndon word and u ≥ v. Then for any nonnegative integer n we have LW(u ∗ un v) = un+1 v. Proof. We now use induction on n. Lemma 3.3.8 takes care of the case n = 0. By the same consideration as in Lemma 3.3.8 we see that LW(u ∗ un v) can only come from either u(un v) = un+1 v or u(u ∗ un−1 v). By induction we are done. Lemma 3.3.10. Let w1 ≥ w2 ≥ · · · ≥ wk be a sequence of Lyndon words. Then LW(w1 ∗ w2 ∗ · · · ∗ wk ) = w1 w2 · · · wk . Proof. We proceed by induction on k. The case k = 1 is trivial. Suppose the theorem is true if the number of Lyndon words is less than k. We then have two cases. 1) If w1 > w2 then w1 > w2 · · · wk since wj ’s are all Lyndon words. By Lemmas 3.3.7, Lemma 3.3.8 and the induction assumption we get LW(w1 ∗ w2 ∗ · · · ∗ wk ) = LW(w1 ∗ w2 · · · wk ) = w1 w2 · · · wk . 2) If w1 = w2 then w2 · · · wk has the form w1n v for v < w1 , and the theorem follows from Lemmas 3.3.7, Lemma 3.3.9 and the induction assumption. Theorem 3.3.11. As a commutative algebra, A∗ is the free polynomial algebra on the Lyndon words. Proof. Let w ∈ A and let w = w1 · · · wk be its Lyndon factorization guaranteed by Lemma 3.3.5. Then by Lemma 3.3.10 we know there is some c ∈ Q such that LW(w − cw1 ∗ · · · ∗ wk ) is smaller than w. Since there are only finitely many words having the same degree as that of w, by repeating this we see that w can be written as a Q-linear combination of the

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stuffle products of non-increasing Lyndon words. Clearly this expression is unique. Let L be the polynomial algebra with one generator for each Lyndon word. The above shows that L maps onto A. Let An be the Q vector space generated by all weight n words in A. We see easily that the Poincar´e series of A is ∞ ∞ X X 1 n . (dimQ An )t = 2n tn = 1 − 2t n=0 n=0 On the other hand, another equivalent definition of a Lyndon word is the following: it is a primitive word (i.e., not a true power of another word) that is minimal in its conjugacy class (i.e., the set of words it produces by moving a right factor to the left). Now if a word w of weight n can be written as w = um with u primitive of weight d then n = dm and the number of conjugates of w is exactly d. Since every weight n word is produced in this way we get X 2n = dN (d). (3.10) d|n

where N (n) is the number of Lyndon words of weight n. Clearly the Poincar´e series of L is ∞ Y f (t) := (1 − tn )−N (n) . n=1

m

The coefficient of t

in the logarithmic derivative (log f (t))0 is X nN (n) = 2m+1 n|(m+1)

by Eq. (3.10). Hence f (t) = 1/(1 − 2t) and therefore L ∼ = A.

Theorem 3.3.12. As a commutative algebra, A
is the free polynomial algebra on the Lyndon words. Proof. The proof is similar to that of Thm. 3.3.11 and can be found in [482, Thm. 6.1], so we leave it to the interested reader. Remark 3.3.13. By M¨ obius inversion formula (see Exercise 3.10) the number of Lyndon words of weight n is given by 1 X n/d N (n) = 2 µ(d), (3.11) n d|n

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where the M¨ obius function  if d = 1;  1, µ(d) = 0, if d has square factor > 1;  (−1)m , if d is the product of m distint primes.

(3.12)

Equation (3.11) can be generalized easily to words built on a larger alphabet. One only need to replace 2 by the number of letters in the alphabet. Theorem 3.3.14. The evaluation maps ζ
and ζ∗ can be extended to the algebra homomorphisms: ζ
:= ζ
(−; T ) : A1
−→ R[T ], where A1
= (A1 ,


),

ζ∗ := ζ∗ (−; T ) : A1∗ −→ R[T ],

A1∗ = (A1 , ∗), and ζ
(x1 ; T ) = ζ∗ (x1 ; T ) = −T.

These provide two ways of regularizations of the MZVs. Remark 3.3.15. We will see that for any non-admissible word w ∈ A1 , both ζ
(w) and ζ∗ (w) are polynomials in T with all coefficients except for the leading one are given by convergent MZVs. To signify this we sometimes write ζ
(w; T ) and ζ∗ (w; T ), which are called regularized MZVs. Proof. By Thm. 3.3.11, as commutative algebras, • A1∗ is the free polynomial algebra on the Lyndon words (with one word x0 removed), • A0∗ is the free polynomial algebra on the Lyndon words (with two words x0 and x1 removed). Thus A1∗ ∼ = A0∗ [x1 ].

(3.13)

So we can naturally extend ζ∗ : A0∗ −→ R to ζ∗ : A1∗ −→ R[T ] by setting ζ∗ (x1 ) = −T . Similarly, by Thm. 3.3.12, as commutative algebras, • A1
is the free polynomial algebra on the Lyndon words (with one word x0 removed), • A0
is the free polynomial algebra on the Lyndon words (with two words x0 and x1 removed).

Thus A1
∼ = A0
[x1 ].

(3.14)

So we can naturally extend ζ
: A0
−→ R to ζ
: A1
−→ R[T ] by setting ζ
(x1 ) = −T .

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Example 3.3.16. For example, x20 x1 x30 x1 =x30 x1 ∗ x20 x1 − x30 x1 x20 x1 + x60 x1 , 1 x20 x1 x20 x1 = (x20 x1 ∗ x20 x1 + x50 x1 ), 2 1 2 x1 x0 = x1 x1 x0 − x1 x0 x1 + x0 x21 . 2





3.3.2




Regularization in Two Ways

Before proving the main theorem for the regularized DBSFs we need a few preliminary results describing how to regularize the divergent series and the divergent iterated integral definition, and how to relate these two kinds of regularizations. Lemma 3.3.17. Define an R-module R-linear automorphism ρ of R[T ] by ∞ X (−1)n ζ(n)un ρ(exp(T u)) = exp n n=2

! exp(T u).

(3.15)

Let P (T ) ∈ R[T ] and Q(T ) = ρ(P (T )). Then      ∞ X 1 1 1 deg P m−1 P (log m + γ)t = Q log + O log 1−t 1−t 1−t m=1 as t → 1+ . Proof. By linearity we only need to prove the identity for P (T ) = (T − γ)l and Q(T ) = ρ((T − γ)l ). First, Weierstrass defined Γ(x) by the following: ∞   u Y 1 u = exp(γu) 1+ exp − . Γ(u + 1) k k k=1

Hence ∞ X ∞  X (−1)n−1 un u u − = γu + . log 1 + k k n kn k=1 n=2 k=1 (3.16) Rearranging the sum we get ! ∞ X (−1)n Λ(u) := exp ζ(n)un = eγu Γ(1 + u) ∀|u| < 1. (3.17) n n=2 P For a power series g(u) = n≥0 an un let Coeff l (g) = al . Then by linearity

− log Γ(u + 1) = γu +

∞  X

Q(T ) = ρ((T − γ)l ) =l! Coeff l ρ(exp((T − γ)u))

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h i =l! Coeff l Λ(u) exp((T − γ)u) i dl h = l Λ(u) exp((T − γ)u) du u=0 i dl h = l Γ(1 + u) exp(T u) du u=0 Thus     1 1 dl Γ(1+u) Q log = l 1−t 1−t du (1−t)1+u u=0 " ∞ # dl X Γ(1+u)(1+u)(2+u) · · · (m−1+u) m−1 = l t du m=1 (m−1)! u=0 " ∞ # l X Γ(m+u) m−1 d = l t du m=1 Γ(m) u=0

∞ X Γ(l) (m) m−1 t . = Γ(m) m=1

Using the following expression (from which one can derive the Stirling’s formula) Z ∞  1 t − [t] − 1/2 log Γ(x) = x − log x − x + C − dt 2 x+t 0 we get Γ0 (x) = Γ(x) log x + Γ(x)O(x−1 )

as x → ∞.

By the Leibnitz rule and the complete induction on l, we have ! l−1 log m Γ(l) (m) = logl m + O as m → ∞, Γ(m) m for all l ≥ 0. By Lemma 2.3.9 we get, as t → 1+ ,   ∞ ∞ X Γ(l) (m) m−1 X 1 l l m−1 t = (log m)t + O log Γ(m) 1−t m=1 m=1   ∞ X 1 l m−1 = P (log m + γ)t + O log , 1−t m=1 as desired. For s ∈ N, define zs = xs−1 x1 . For any composition s = (s1 , . . . , sd ) ∈ 0 d N , we set zs := zs1 · · · zsd and ζ∗ (s; T ) := (−1)d ζ∗ (zs ),

ζ
(s; T ) := (−1)d ζ
(zs ).

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Remark 3.3.18. We will always use ζ
(s; T ) and ζ∗ (s; T ) when we consider them as polynomials of T , while we will write ζ
(s) := ζ
(s; 0) and ζ∗ (s) := ζ∗ (s; 0) for the constant terms only. For M > 0, we define the partial sum of the MZVs X HM (s) := k1−s1 · · · kd−sd .

(3.18)

M ≥k1 >···>kd >0

It is easy to see that (−1)d+` HM (s)HM (t) = HM (zs ∗ zt ) where HM is the Q-linear map A1∗ −→ R such that HM (zs ) = (−1)d HM (s). Lemma 3.3.19. Put J = dp(s). Then as M → ∞ we have HM (s) = ζ∗ (s; log M + γ) + O(M −1 logJ M ).

Moreover, if s = ({1}` , s0 ) for admissible s0 then the leading term of ζ∗ (s; T ) is (1/`!)ζ(s0 )T ` , where ζ(∅) = 1. Proof. We first prove that for all d ≥ 1 and j ≤ d we have X 1 = O(M −1 logd M ). Aj,d (M ) = 2 k1 k2 · · · kd k1 >···>kj >M ≥kj−1 >···>kd >0

(3.19)

Indeed, if d = 1 we have j = 1 and X 1 1 X 1 < = O(M −1 log M ). A1,1 (M ) = 2 k M k k>M

k>M

Now by induction X k1 >···>kj >M ≥kj−1 >···>kd >0



X k1 >···>kj >M

1 k12 k2 · · · kj

···

1 k12 k2 · · · kd X

M ≥kj−1 >···>kd−1 >0

 d−1  log kd−1 log M +O kj+1 · · · kd−1 M

 d−1  X logj−1 k1 log M + O k12 M k1 >M  d−1  Z ∞ d−j log M logj−1 x log M logd−1 M  dx + O  . M x M M M (logd−j M )

This proves Eq. (3.19).

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We now proceed by induction on the number ` of leading 1’s in s. If ` = 0 we have s1 > 1. ζ(s) − HM (s) = ≤

d X

X

j=1 k1 >···>kj >M ≥kj−1 >···>kd >0 d X

k1−s1 · · · kd−sd

X

j=1 k1 >···>kj >M ≥kj−1 >···>kd >0

 d−1  log M 1 = O k12 k2 · · · kd M

by Eq. (3.19). This proves the case ` = 0 with ζ∗ (s; T ) = ζ(s) a constant. Suppose the lemma is true when the number of leading 1’s in s is less than ` ≥ 1. Suppose further that s = ({1}`−1 , t1 , . . . , td ) with t1 > 1 for some d ≥ 0 (when d = 0 no tj ’s will appear). Then −HM (x1 ) = HM (1) =

M X 1 = log M + γ + O(M −1 ) k

k=1

as M → ∞,

where γ is Euler’s constant. Using the stuffle relation we have `HM (1, s) = HM (1)HM (s) − − −

`−1 X j=1 d X

d X

HM ({1}`−1 , t1 , . . . , tj , 1, tj+1 , . . . , td )

j=1

HM ({1}j−1 , 2, {1}`−j−1 , t1 , . . . , td ) HM ({1}`−1 , t1 , . . . , tj−1 , tj + 1, tj+1 , . . . , td ).

j=1

By the induction assumption `HM (1, s) logJ+1 M = (log M +γ)ζ∗ (s; log M +γ)+O M 

− − −

d X

`−1

ζ∗ ({1}

j=1 `−1 X

j−1

ζ∗ ({1}

j=1 d X j=1

`−1

ζ∗ ({1}



logJ+1 M , t1 , . . . , tj , 1, tj+1 , . . . , td ); log M +γ)+O M 

, 2, {1}

`−j−1

logJ M , t1 , . . . , td ; log M +γ)+O M 





logJ M , t1 , . . . , tj−1 , tj +1, tj+1 , . . . , td ; log M +γ)+O M

= `ζ∗ (1, s; log M +γ)+O(M −1 logJ+1 M ),





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since ζ∗ (1; T )ζ∗ (s; T ) = T ζ∗ (s; T ) can be computed by the stuffle, which leads to the expression of ζ∗ (1, s; T ). In particular, the leading term of ζ∗ (1, s; T ) is equal to T` times the leading term of ζ∗ (s; T ). Thus the lemma follows from the inductive assumption. Now for |z| < 1 we define the single-variable MPL by Z z X z k1 d x0s1 −1 x1 · · · xs0d −1 x1 , Ls (z) := = (−1) k1s1 · · · kdsd 0 where x0 =

(3.20)

k1 >···>kd ≥1 dx dx x and x1 = x−1 .

Lemma 3.3.20. Put J = dp(s). Then as t → 1+      1 1 J Ls (t) = ζ
s; log + O (1 − t) log . 1−t 1−t

(3.21)

Proof. We prove the lemma by induction on the number ` of leading 1’s in s. First assume ` = 0 and so s1 > 1. Set paths αt = [0, t] and βt = [t, 1]. By Lemma 2.1.2(iii) we have X 1 − tk1 Ls (1) − Ls (t) = s1 k1 · · · kdsd k1 >···>kd ≥1

≤

X k1 >···>kd ≥1

1 − tk1 k12 k1 · · · kd

= L(2,{1}d−1 ) (1) − L(2,{1}d−1 ) (t)  j−1 Z t  d−j d Z 1 X dt dt dt = t 1−t 1−t 0 j=1 t  d−1 R1 dt . Now by applying Lemma 2.1.2(ii) to L(2,{1}d−1 ) (1) = 0 dt t 1−t d−j   Z t dt 1 = logd−j 1−t 1−t 0 and by the change of variables tj → 1 − tj and Lemma 2.1.2(iv)  j−1 Z 1−t  j−1 Z 1 dt dt dt dt − = = Lj (1 − t) = O(1 − t). t t − 1 t 1 −t t 0 Thus    1 Ls (t) = ζ(s) + O (1 − t) logd . 1−t Since ζ
(s; T ) = ζ(s) this proves the lemma when ` = 0. The induction step can now be finished by using the shuffle relation, similar to the proof of Lemma 3.3.19. We leave out the details. This finishes the proof of the lemma.

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3.3.3

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Main Statement

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The following result provides the regularized double shuffle relations (regularized DBSFs). Theorem 3.3.21. For all w ∈ A1 , one has

(ζ
− ρ ◦ ζ∗ )(w) = 0.

Proof. Let J = dp(s). Notice ∞ X Ls (t) =

X

(3.22)

tm

ms1 k2s2 · · · kdsd m=1 m>k2 >···>kd ≥1 ∞ X = (Hm (s) − Hm−1 (s))tm m=1 ∞ X Hm (s)tm = (1 − t) m=1

since H0 (s) = 0. Thus      1 1 Lem. 3.3.20 J ζ
s; log ======= Ls (t) + O (1 − t) log 1−t 1−t ! ∞ ∞ X X logJ m m−1 Lem. 3.3.19 m ζ∗ (s; log m + γ)t + O (1 − t) ======= (1 − t) t m m=1 m=1    ∞ X 1 Lem. 2.3.9 J+1 m ζ∗ (s; log m + γ)t + O (1 − t) log ======= (1 − t) 1−t m=1      1 1 Lem. 3.3.17 + O (1 − t) logJ+1 ======= tQ log 1−t 1−t      1 1 t=1+(t−1) ======= Q log + O (1 − t) logJ+1 1−t 1−t with Q(T ) = ρ(ζ∗ (s; T )) since degT ζ∗ (s; T ) ≤ J. Thus ζ
(s; T ) = ρ(ζ∗ (s; T )). The theorem is now proved since s ∈ Nd is arbitrary. Remark 3.3.22. One can compute ρ explicitly as follows. First expand Λ(u) into a power series: ! ∞ ∞ X X (−1)n n Λ(u) = exp ζ(n)u = ck uk , n n=2 k=0

where c0 = 1, c1 = 0, c2 = ζ(2)/2, c3 = −ζ(3)/3, c4 = 9ζ(4)/16, · · · . Then   n X n n−k n ρ(T ) = k!ck T , ∀n ≥ 0. k k=0

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Corollary 3.3.23. For all w0 ∈ A0 and w1 ∈ A1 , one has

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ζ
(w1


w

0

− w1 ∗ w0 ) = 0.

(3.23)

Proof. We can get the desired result by multiplying ζ(w0 ) on both sides of ζ
(w1 ) = ρ(ζ∗ (w1 )) and using R-linearity of ρ. Definition 3.3.24. The regularized double shuffle relations (regularized DBSFs) of the MZVs are those Q-linear relations obtained by setting T = 0 in Eq. (3.22) for all words w ∈ A1 . Conjecture 3.3.25. The non-finite regularized DBSFs of the MZVs are generated by the Hoffman relations ζ(x1 w − x1 ∗ w) for all admissible words w.




If w ∈ A1 \ A0 then the leading letter of w must be x1 . The conjecture says that to produce all the non-finite regularized shuffle relations we only need to regularize divergent MZVs with only one leading x1 . Example 3.3.26. The simplest regularized DBSF is obtained by considering the word x1 x0 x1 ∈ A1 . First, since ζ(2) = −ζ
(x0 x1 ) and T = −ζ
(x1 ) ζ(2)T = ζ
(x0 x1


x ) = ζ
(x x x ) + 2ζ
(x x ) = ζ
(x x x ) + 2ζ(2, 1). 1

1 0 1

2 0 1

1 0 1

On the other hand, ζ(2)T = ζ∗ (x0 x1 )ζ∗ (x1 ) = ζ∗ (x0 x1 ∗ x1 ) = ζ∗ (x1 x0 x1 ) − ζ∗ (x20 x1 ) + ζ∗ (x0 x21 ). Observe that −ζ∗ (x0 x0 x1 ) = ζ(3) = ζ(2, 1) = ζ∗ (x0 x1 x1 ) by (3.24). Applying Thm. 3.3.21 to w = x1 x0 x1 we get ζ(2)T − 2ζ(2, 1) = ρ(ζ(2)T − ζ(3) − ζ(2, 1)) = ζ(2)T − ζ(3) − ζ(2, 1) since ρ(T ) = T by Eq. (3.15). This leads to the famous identity first discovered by Euler: ζ(3) = ζ(2, 1).

(3.24)

3.4 Dimension Conjectures Over Q One of the guiding principles in the study of the MZVs is the following widely believed conjecture. Conjecture 3.4.1. There are no nontrivial Q-linear relations among the MZVs of different weights.

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Multiple Zeta Values (MZVs)

This means that every MZV identity involving different weights is a sum of a bunch of identities each of which has pure weight. For the MZVs of the same weight, by Example 3.3.2 and Example 3.3.26, often there exist Q-linear relations. Let MZw be the Q-vector space spanned by all the MZVs of weight w. Zagier made the following famous conjecture. Conjecture 3.4.2. Define the sequence Dw (w ≥ 1) by D0 = 1, D1 = 0, D2 = 1, Dw = Dw−2 + Dw−3 ∀w ≥ 3.

(3.25)

Then dim MZw = Dw for all w ≥ 1. Remark 3.4.3. The numbers Dw are called Padovan numbers. According to OEIS [518] the Padovan sequence A000931 is 1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, . . . . Using the finite and regularized DBSFs one can show that each MZV of weight less than 8 is a polynomial of ordinary Riemann zeta values with Q coefficients. But already ζ(3, 5) of weight 8 seems to fail this rule even if no proof has been found. In fact, this would be the case if the following conjecture could be verified. Conjecture 3.4.4. Every linear relation among the MZVs can be proved by the finite and regularized DBSFs. It is quite fruitful to reinterpret some of the above results in terms of a certain graded Hopf algebra (see Appendix A). Let L denote the free Lie algebra generated by one element e2n+1 in every odd degree −2n − 1 for n > 1. Graded by the degree the underlying Q-vector space is generated by, e3 ;

e5 ;

e7 ;

[e3 , e5 ];

e9 ;

[e3 , e7 ];

e11 , [e3 , [e5 , e3 ]];

··· .

Let UL be its universal enveloping algebra, and let UL∨ be its graded dual, which is a commutative Hopf algebra. One may think of UL∨ as the set of all non-commutative words in letters f2n+1 in degree 2n + 1 dual to e2n+1 , equipped with the shuffle product . Finally, set




A = Q[f2 ] ⊗ Q(hf3 , f5 , f7 , . . . i,


)

where f2 is a new generator of degree 2 which commutes with all the others. The generators in each weight w with w ≤ 9 are f2 ;

f3 ;

f22 ;

f24 , f2 f32 , f3

f5 , f2 f3 ;


f ,f f ; 5

3 5

f23 , f32 ;

f7 , f2 f5 , f22 f3 ;

f23 f3 , f33 , f2 f7 , f22 f5 , f9

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which correspond to the following MZVs: ζ(2); ζ(3); ζ(2)2 ; ζ(5), ζ(2)ζ(3); ζ(2)3 , ζ(3, 3); ζ(7), ζ(2)ζ(5), ζ(2)2 ζ(3); Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 06/12/16. For personal use only.

ζ(2)4 , ζ(2)ζ(3, 3), ζ(3)ζ(5), ζ(3, 5); ζ(2)3 ζ(3), ζ(3)3 , ζ(2)ζ(7), ζ(2)2 ζ(5), ζ(9) The next conjecture is a more precise version of Zagier’s Conjecture 3.4.2 . Conjecture 3.4.5. The algebra spanned by the MZVs over Q is isomorphic to A. Actually, we only need the elements f2 and f3 to generate all the others by the rule f2n+1 → f3 f2n−1

∀n ≥ 1.

Hence if we denote F(2, 3) the free graded Lie algebra generated by two elements of degree −2 and −3, and F(2, 3)∨ its graded dual, then with the same notation we have a canonical isomorphism of graded vector spaces A∼ = UF(2, 3)∨ . Let d0w be the dimension of the graded w piece of A. Then it is clear that d01 = 0, d02 = 1, d03 = 1, d0w = d0w−2 + d0w−3 ∀w ≥ 4. One can even consider MZVs with both depth and weight fixed. The next conjecture provides the hypothetical dimensions of Q-spaces generated by such values. Let MZw,≤d be the Q-vector space spanned by all the MZVs of weight w and depth at most d. Let X MZw,+,≤d := MZw1 ,≤d1 · MZw2 ,≤d2 w1 ,w2 ≥1, w1 +w2 =w 1≤d1 ,d2 t1 >···>tw >0

where the integrand is the same as in the ordinary MZV defined by Eq. (3.1). Now applying the formula Z Z dx1 a r dxr 1  ··· log ··· = x1 xr r! b a>x1 >···>xr >b

to (x0 , b, r) = (1 − ε, u1 , s1 − 1), (u1 , u2 , s2 − 1), . . . , (un−1 , un , sn − 1), where u1 = ts1 , u2 = ts1 +s2 , . . . , un = tw , we find 1 ζε (s) = (s1 − 1)! · · · (sn − 1)! 

Easy computation using F
,ε (x1 , . . . , xn ) := (n)

u1 log u2 P i≥0

Z

···



Z

1−ε>u1 >···>un >0

s2 −1

1−ε log u1

s1 −1

du1 1 − u1

 s −1 du2 un−1 n dun · · · log . 1 − u2 un 1 − un

tn /n! = et yields

X

ζε (s1 , . . . , sn )x1s1 −1 . . . xsnn −1

s1 ,...,sn ≥1

= (1 − ε)x1

Z

···

−x

Z

1−ε>u1 >···>un >0

+x

n u n−1 n dun−1 u−x dun u1−x1 +x2 du1 n · · · n−1 . 1 − u1 1 − un−1 1 − un

Thus (F
,ε )] (x1 , . . . , xn ) = F
,ε (x1 + · · · + xn , . . . , xn−1 + xn , xn ) Z Z −x 1 n u n−1 dun−1 u−x dun u−x du1 n 1 = (1 − ε)x1 +···+xn ··· · · · n−1 . 1 − u1 1 − un−1 1 − un (n)

(n)

1−ε>u1 >···>un >0

Hence (F
,ε )] satisfies the shuffle relation (n)


(j) (n−j) (n) (F
,ε )] (x1 , . . . , xj )(F
,ε )] (xj+1 , . . . , xn ) = (F
,ε )] | shj (x1 , . . . , xn ) for any ε < 1 and 1 ≤ j < n. The lemma now follows from the definition of ζ
(s; T ).

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Theorem 3.5.3. For all n ≥ 1, we have

F
(x1 , . . . , xn ) = F∗ (x1 , . . . , xn ) (3.29) P in M ⊗Q Vn . Moreover, both belong to the subspace w MZw,≤n ⊗Q DShn P modulo w MZw,+,≤n .

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(n)

(n)

Proof. First, we establish Eq. (3.29). If n = 1 we see by the definition that ζ
(s) = ζ∗ (s) = (1 − δ1,s )ζ(s) for all positive integer s, where δ1,s (1) is the Kronecker symbol. Thus the two generating functions F
(x1 ) and (1) F∗ (x1 ) are exactly the same. When n ≥ 2 by Thm. 3.3.21, we see that each (n) (n) coefficient of F
(x1 , . . . , xn ) − F∗ (x1 , . . . , xn ) comes from the products of some MZV factors with the coefficients of ρ(T i ). Now the coefficients of ρ(T i ) are all Q-linear combinations of at most i Riemann zeta values (see Eq. (3.15)) which can be written as Q-linear combinations of the MZVs of depth at most i. Then MZV factors can be easily seen to have depths at most n − i (for each T the weight of the corresponding MZVs must be decreased by 1). To prove the second sentence in the theorem, we have (j)

(n−j)

0 = F∗ (x1 , . . . , xj )F∗ (xj+1 , . . . , xn ) X = ζ∗ (s1 , . . . , sj )ζ∗ (sj+1 , . . . , sn )x1s1 −1 . . . xsnn −1 s1 ,...,sn ≥1

=

X

X

ζ∗ (sσ−1 (1) , . . . , sσ−1 (n) )x1s1 −1 . . . xsnn −1

s1 ,...,sn ≥1 σ∈Sh(n) j

(n) = (F∗ | shj )(x1 , . . . , xn ),

since the “stuffing” part of the stuffle product always produces the MZVs of lower depth. By Eq. (3.29) and Lemma 3.5.2 the theorem follows immediately. The following bound on the dimension of MZw,d (see Sec. 3.4) is an easy consequence of Thm. 3.5.3. The proof is left as an exercise (see Exercise 3.14). Corollary 3.5.4. For all w > d > 0, we have Dw,d ≤ dimQ DShd (w − d). Our next goal is to prove the Parity Principle in Cor. 3.5.7. The idea is to define a suitable space ShCn (d) containing the double shuffle space DShn (d) and satisfying ShCn (d) = {0} when d is odd and n ≥ 1. In order

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Multiple Zeta Values (MZVs)

to do so we need to define an action of Sn+1 on Vn (extending that of Sn ) by

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(F |σ)(x1 , . . . , xn ) := F (xσ−1 (1) , . . . , xσ−1 (n) ),

for all σ ∈ Sn+1 ,

where xn+1 = −x1 − x2 − · · · − xn . Let   1 ··· n − 1 n n + 1 Cn+1 = ∈ Sn+1 . 2 ··· n n + 1 1

(3.30)

Then (f |Cn+1 )(x1 , . . . , xn ) = f (xn+1 , x1 , . . . , xn−1 ) so that the matrix form of Cn+1 in GLn (Z) is     −1 1 0 · · · 0 −1   .. 1 −1 −1 0  .   −1  . Cn+1 =  . ..  , and Cn+1 =  .  . . .  .. .. . .   .. 1  1 −1 −1 0 ··· 0

Let ν(xj ) = −xj for all j ≤ n then its matrix form in GLn (Z) is −In where In is the n × n identity matrix. Clearly ν(xn+1 ) = −xn+1 . Lemma 3.5.5. For 0 ≤ j ≤ n, define the involutions   1 ··· j j + 1 ··· n Tj = ∈ Sn . 1 ··· j n ··· j + 1

(3.31)

Then

n−1 X

(−1)j−1 shj Tj = T0 + (−1)n .

j=1

Proof. For 1 ≤ j ≤ n, we define Rj =

X

σ.

σ∈Sn ,σ(1)···>σ(n)

Then sh0 T0 = R1 , shn Tn = Rn , and for 1 ≤ j ≤ n − 1, we have X shj Tj = σ = Rj + Rj+1 , σ∈Sn ,σ(1)σ(n)

since either σ(j) > σ(j + 1) or σ(j) < σ(j + 1). The lemma follows immediately. Theorem 3.5.6. For n, d ≥ 1, we define n o ShCn (d) := f ∈ Vn (d) f ] | shj = 0 (1 ≤ j < n), f ] |Cn+1 = ν(f ] ) .

(3.32)

Then

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(i) DShn (d) ⊂ ShCn (d); (ii) ShCn (d) = 0 if d is odd. Proof. (i) By Lemma 3.5.5, for all f ∈ DShn (d) we have f |T0 = f |T0] = (−1)n−1 f.

(3.33)

Define the involution   1 2 ··· n − 1 n n + 1 0 T = ∈ Sn+1 . n − 1 n − 2 ··· 1 n + 1 n

(3.34)

Then (f |T 0 )(x1 , . . . , xn ) = f (xn−1 , . . . , x1 , xn+1 ) so that the matrix form of T 0 in GLn (Z) is   1 −1  ..   .·· .  T 0 = T 0−1 =  . 1 −1 0

···

0

−1

It is not hard to use the matrix forms to show that (see also Excercise 3.15) (A). T0 T 0 = Cn+1 ,

(B). P T 0 = νT0 P.

and

(3.35)

Thus (3.35)(A)

defn

defn

f ] |Cn+1 ===== f ] |T0 T 0 == (f |P )|T0 T 0 = (f |P T0 )|T 0 == (f |T0] P )|T 0

(3.33)

(3.35)(B)

(3.33)

=== (−1)n−1 (f |P )|T 0 ===== (−1)n−1 ν(f |T0 P ) === ν(f |P ) = ν(f ] ).

So f ∈ ShCn (d) and therefore (i) is proved. (ii) Let τ1,n+1 be the transposition 1 ↔ n + 1. It is straightforward to check (see Exercise 3.16) that in Z[Sn+1 ] (n)

(n)

1 + sh1 Cn+1 = Cn+1 (1 + sh1 τ1,n+1 ).

(3.36)

]

Now suppose f ∈ ShCn (d) and let F = f . Then F | sh1 = 0 and F |Cn+1 = νF by the definition. Thus by Eq. (3.36)
(n) (n) F = F |Cn+1 (1 + sh1 τ1,n+1 ) = ν F |(1 + sh1 τ1,n+1 ) = νF = −F

if the degree d is odd. Hence F = 0 and thus f = F |P −1 = 0. This completes the proof of the theorem. Corollary 3.5.7. (Parity Principle) If the weight w and depth n of ζ(s) have different parities then ζ(s) is a Q-linear combination of lower depth MZVs and products of lower weight MZVs whose sum of depths is bounded by that of s.

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Proof. If |s| and n have different parities then the degree of x1s1 −1 . . . xsnn −1 is odd. So the corollary follows from part (ii) of the above theorem.

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The next result is another kind of Parity Principle. Corollary 3.5.8. If the w = |s| and n have the same parity then both ← ← ζ∗ (s) + (−1)w ζ∗ ( s ) and ζ
(s) + (−1)w ζ
( s ) are Q-linear combinations of lower depth MZVs and products of lower weight MZVs. Proof. By Eq. (3.33) we see that X X ← ← (n) (n) ζ∗ (s)x| s |−1 = (F∗ |T0 )(x) = (−1)n−1 F∗ (x). ζ∗ ( s )x|s|−1 = s∈Nn

s∈Nn

So we get ζ∗ (s) + (−1)w ζ∗ ( s ) ≡ ζ∗ (s) + (−1)w+n−1 ζ∗ (s) ≡ 0 ←

modulo MZw,+,≤n . Similar argument is valid for the completes the proof of the corollary.


-version, too. This

3.6 Historical Notes Even though some special cases of the theory of the multiple zeta values were discovered by Goldbach and Euler, their formal definition and study was initiated only recently by Hoffman [288] and Zagier [598] independently. Incidentally, in early 1980s, while developing a theory of resurgence in complex analysis, Ecalle studied some quite general mathematical objects called “moulds” (functions with variable number of variables) of which the MZVs are one of the examples [185,186, p. 429]. He even mentioned their iterated integral representation [185, p. 431, Remark 4] without explicitly producing it. For an introduction to this theory, see Schneps’ exposition [502]. In recent years, the study of the MZVs has opened up quite a lot fascinating connections to other branches of mathematics such as knot theory, Mahler measures and the theory of motives (see [83, 102, 104, 164, 368]). MZVs have also emerged unexpectedly in high energy physics such as Feyman diagrams, amplitude, mirror symmetry and quantum field theory (see Chap. 15 and Refs. [83, 90, 108, 291, 500]). One of the most important algebraic approaches to the study of the MZVs is through the connection between the two kinds of products: shuffle and stuffle, namely, the regularized DBSFs. The name “stuffle” first appeared in [71, 72] although such algebras appeared implicitly already in Cartier’s work [116] in early 1970s. It is also called quasi-shuffle in [290] using recursive definition, harmonic shuffle in [541], or sticky shuffle in [302].

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It is equivalent to the algebra structure called mixable shuffle treated in [263] which generalizes the shuffle product in terms of permutations. In a series of papers [289–291], Hoffman first realized that one may equip the MZVs with two different algebra structures reflecting the two different ways to multiply the MZVs. This has become one of the cornerstones in various studies of the MZVs and their generalizations. It turns out the MZV relations obtained by the shuffle product of the iterated integrals can often be proved using other methods. We have derived Euler’s decomposition formula (i.e., Thm. 3.1.1) by using the shuffle product. In [191], Eie extended this in several ways by varying the binomial coefficients with carefully chosen integrals. However, Euler originally discovered his formula with partial fractions (see the proof of Thm. 5.3.1). This is important because the integral representation is not always easily available in some generalizations of the MZVs, but the partial fraction technique can still be applied. For example, Thm. 3.1.1 were enhanced to a certain q-analogs (see Chap. 11) by the author [615] with the iterated Jackson q-integrals in a very complicated manner. But by using partial fractions, Bradley [79] found a much cleaner form with a shorter proof. Another q-analog of Euler’s decomposition formula can be found in [119]. This formula is later generalized to the product of any two MZVs by the algebraic method of double shuffle product by Guo and Xie [262]. With different approaches, it is also extended to the cases involving the MZVs of the form ζ(m, {1}n ) in [193, 265]. Moreover, in [261] Guo and Xie showed that the class of fractions produced from MZVs has a natural shuffle algebra structure which explains why the partial fractions can be used as surrogates of the iterated integrals. On the other hand, the stuffle product leads to so-called Delannoy numbers (see Exercise 3.1). K.-W. Chen [136] derived some interesting relations among these numbers by using the stuffle structure of the MVZs. From Euler’s decomposition formula, he and his collaborators Chung and Eie [138] also produced several combinatorial identities extending that of Pascal. Incidentally, both the shuffle and stuffle structures appear unexpectedly in a few other fields in recent years. For instance, in [155] Curry et al. used this to study the algebra of repeated integrals of semimartingales and proved that a finite family of independent L´evy processes that have finite moments generates a minimal family. In real world applications, efficient simulation of L´evy processes have been regarded as one of the key steps in finance, economics and insurance modeling. Proposition 3.2.6 appears first in [71]. We will use it to establish some

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MZV identity in Theorem 5.8.1. A similar result due to Yee [587] is given in Exercise 3.3. Lemma-Definition 3.3.5 is a special case of Chen–Fox–Lyndon Theorem since it can be derived from [134], but the first explicit statement for the algebra of words representing the MZVs did not appear until the publication of [506], although Lemma 3.3.7 and a few other results in Sec. 3.3 can be found in [289, 482]. For Remark 3.3.13 on the classical M¨obius Inversion Formula, see [404, (1.3.7)]. Theorem 3.3.21 giving the regularized DBSFs was stated by Ihara, Kaneko and Zagier in Ref. [308] parts of which had circulated as preprints long before its formal publication. Incidentally, the regularized DBSFs are called the extended DBSFs in [308]. In the same paper they also considered the derivation relations which we will discuss in Chap. 5. Z. Li [395] treated some variation of the shuffle regularization using the single-variable MPLs and the Drinfeld associator. In [338] Kaneko et al. conjectured that a certain concrete subset of the regularized DBSFs can generate all the regularized DBSFs which was verified by Combariza [145] with some numerical data. Note that the regularized DBSFs themselves are not explicit formulas of the MZVs, neither are the finite ones. In [262] Guo and Xie described an idea of how to write down these explicitly and carried this out for products of two double zeta values. Hoang Ngoc et al. further conjectured in [285] that all the Q-linear relations among the MZVs can be produced by the finite DBSFs plus those of the form ζ(u ∗ x1 − u x1 ) = 0 with admissible u. Note also that by emulating Tate’s idea Horozov [299] gave an interpretation of the MZV DBSFs using iterated integrals over the finite ideles and over the idele class group. As he remarked, this idea might be modified to handle the DBSFs of function field MZVs which are currently missing. The folklore Conjecture 3.4.1 is a consequence of some deep conjectures in the theory of algebraic geometry and motives (see [246, Conjecture B]). Zagier further proposed Conjecture 3.4.4 while Conjecture 3.3.25 is a much stronger form of it. Clearly, Conjecture 3.4.5 also implies the next.




Conjecture 3.6.1. (Hoffman’s Basis Conjecture) The Q-space generated by the MZVs has a basis consisting of the MZVs whose arguments are equal to either 2 or 3 (such MZVs are called the Hoffman elements). A deep result of Brown [102] (see also [162] for an exposition) implies that every MZV is a Q-linear combination of the Hoffman elements. Proposition 3.2.5 is due to Borwein, Bradley and Broadhurst (see [71, Prop. 1]). The idea to use relations between the words in the Hoffman

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algebra is one of the most powerful tools to investigate the Q-linear relations among the MZVs. Zagier’s Dimension Conjecture 3.4.2 has been one of the driving forces in the study of the MZVs. Its motivic version, Conjecture 3.4.5, was stated in [250]. Since Conjecture 3.4.2 is an equality one may approach it by verifying two inequalities. Goncharov and Terasoma (see [250, 540]) independently proved that dimQ MZw ≤ Dw . This inequality also follows from Brown’s recent result mentioned above, which confirms only half of Hoffman’s Basis Conjecture. To prove the other half of Conjecture 3.4.2 one now needs to show dimQ MZw ≥ Dw which can be deduced from some notoriously difficult conjectures in transcendental number theory (more precisely, the Grothendieck’s period conjecture, see [164, § 5.27] for details). For example, we know π is irrational by Lampert’s work in [374] of 1768 (and transcendental by Lindemann [401] more than a century later). Almost another century had passed before Ap´ery [19] proved ζ(3) is irrational in 1978. An additional twenty plus years later, Ball and Rivoal [35, 484] showed that infinitely many zeta values ζ(2n + 1) are irrational. More recently, Zudilin [642] and others made some remarkable refinements in this direction. The progress seems to be quite slow, and in fact, until today we don’t even know whether ζ(5) is irrational or not. In earlier 1990, Bailey et al. already noticed, on the basis of extensive numerical experiments, that some double zeta values such as ζ(6, 2) cannot be expressed as a Q-linear combination of products of the Riemann zeta values (assuming ζ(0) = −1/2 is also a Riemann zeta value). To detect the fine structure of the MZVs, Broadhurst and his collaborators computed extensively with the aid of computers and observed a lot of subtle properties. In [83] he gave the precise Conjecture 3.4.6 for the number of irreducible MZVs of given weight and depth just by observing the patterns of the known dimensions. This formula has since been explained partly by a hidden relation between the MZVs and the modular forms (see [233] and Sec. 5.9). Note that each of the three conjectural formulas in Eq. (3.26) is known to give an upper bound for the corresponding true dimension (see [597] for d = 2 and [249] for d = 3 or [305] by a different method). The ideas from [305] and some homological computations involving finite subgroups of GL4 (Z) by Yasuda [586] may be useful to treat the d = 4 case. In the general case, Carr, Gangl and Schneps have found some convincing evidence in [115]. The Q-algebra generated by the MZVs and the algebra F(2, 3) are

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Multiple Zeta Values (MZVs)

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closely related to some stable derivation algebra coming from braid groups (see [159, 173, 215, 309–311, 552]). This provides the key links Brown used in [102] (see also [162]) to prove the Deligne-Ihara conjecture on the outer action of Gal(Q/Q) on the pro-fundamental group of P1 \ {0, 1, ∞}. For the study of the MZVs from the algebro-geometric point of view, one can consult [221, 223, 521–523]. Tsumura’s original proof in [550] of the Parity Principle (Cor. 3.5.7) contains heavy analysis. An algebraic proof was later given by Ihara, Kaneko and Zagier (see [308, Thm. 7]) which we have modified and adopted in Sec. 3.5. Note that the case d = 2 was already known to Tornheim [547] and the case d = 3 was proved by Borwein and Girgensohn [69]. For lower depth cases it is possible to explicitly write down the reduction formulas, for example, see [118, 460, 461, 601]. Similar parity result for other kind of the multiple zeta functions should exist. For example, Okamoto [452] discovered such a result for the Witten zeta function ζ2 (s; G2 ) for Lie algebra G2 in some extended sense. There are a variety of ways to generalize multiple zeta functions and their special values, such as those of the Hurwitz-type [8, 75, 75, 170, 322, 345,351,418,509,634]. In particular, if the parameters in these Hurwitz-type special values are rational numbers with denominators equal to a divisor of a fixed positive integer N , then they become essentially so-called MZVs of level N . For pertinent research along this line, see [334, 438, 590, 591]. For general surveys of the results on the MZVs up to the beginning of twenty-first century, the interested reader may peruse Refs. [117, 563, 643] by Cartier, Waldschmidt and Zudilin, respectively. Exercises 3.1. Let u and v be two
words. Let a = dp(u) and b = dp(v). Show that dp(u v) = a+b and dp(u ∗ v) = D(a, b) is the Delannoy number a (see [146, p. 81]) that satisfies D(a, 0) = 1 for all a ≥ 0 and




D(a, b) = D(a, b − 1) + D(a − 1, b) + D(a − 1, b − 1) ∀a, b ≥ 1. j 3.2. Let Tp+q be defined as in Prop. 3.2.5. Show that the factor x21 appears exactly j times.

3.3. In the word (x40 x21 )n , there are 2n−1 ways in which one can exchange a
letter x0 with an adjacent letter x1 . Let Ξn (k) denote the sum of the 2n−1 k

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words obtained from (x40 x21 )n by making k such transpositions. Show that n X

h (−1)r (x20 x1 )n−r


(x x )

2 n+r 0 1

i

= 3n

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r=−n

2n−1 X

22n−k Ξn (k).

k=0

3.4. Prove Prop. 3.2.8 by induction. 3.5. Complete the proof of Lemma 3.3.7. 3.6. Complete the proof of Thm. 3.3.12. 3.7. Find the finite DBSF by computing ζ(2)ζ(3) in two different ways. 3.8. Using Definition 3.3.3 how many Lyndon words of weight six can you find in Qhx0 , x1 i? 3.9. (1). Find the Lyndon factorization of x21 x20 x1 and x1 x30 x31 x50 x21 x20 x1 x50 x1 x30 . (2). Express each word in (1) as a Q-linear combination of ∗-products of Lyndon words. (3). Express each word in (1) as a Q-linear combination of -products of Lyndon words. P 3.10. Let f and g be two real functions defined on N. If g(n) = d|n f (d) P for all n ∈ N, prove that f (n) = d|n f (d)µ(n/d) for all n ∈ N where µ is the M¨ obius function defined by Eq. (3.12). This is called the M¨obius inversion formula.




3.11. Find the regularized DBSF by computing ζ(3)T in two different ways using Thm. 3.3.21 and Remark 3.3.22. 3.12. By hand computation verify Zagier’s Dimension Conjectuer 3.4.2 provides the upper bound for dimQ MZw for w ≤ 5. 3.13. This exercise is for Maple users. First, download the software mzv16.m (or mzv12.txt if your computer has moderate RAM) from Petitot’s website http://www.lifl.fr/ petitot/recherche/MZV/. Then • Verify that Broadhurst Conjecture 3.4.6 provides the upper bound for dimQ MZw,d for d < w ≤ 12. • Verify the Parity Principle (Cor. 3.5.7) for all MZVs of weight 10. 3.14. Prove Cor. 3.5.4.

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Multiple Zeta Values (MZVs)

3.15. Recall that P , T 0 and T0 are defined by Eqs. (3.28), (3.34) and (3.31), respectively. Show that for all f ∈ Vn Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 06/12/16. For personal use only.

((f |P )T 0 )(x1 , . . . , xn ) = (f |T0 P )(x1 , . . . , xn )

= f (−xn , −xn − xn−1 , . . . , −xn − · · · − x1 ). (n)

3.16. Recall that τ1,n+1 is the transposition 1 ↔ n + 1, and sh1 and Cn+1 are defined by Eq. (3.32) and Eq. (3.30), respectively. Show that by regarding a1 . . . an+1 as a word (n)

(1 + sh1 Cn+1 )(a1 . . . an+1 ) (n)

= Cn+1 (1 + sh1 τ1n+1 )(a1 . . . an+1 ) = a1 . . . an+1 + an+1 a1 . . . an + a1 [an+1 = an+1


(a

1


(a

2

. . . an−1 )]an

. . . an ).

3.17. For any w, d ∈ N, show that ζ ? (w, {1}d−1 ) =

d X

X

ζ(ct−1 + w, ct−2 + 1, . . . , c1 + 1).

t=1 c1 +···+ct−1 =d cj ≥0∀j=1,...,t−1

3.18. Suppose f ∈ Q[x1 , x2 ] has homogeneous degree d. Show that f ∈ DSh2 (d) if and only f (x1 , x2 ) + f (x2 , x1 ) = 0

and

f (x1 + x2 , x1 ) + f (x1 + x2 , x2 ) = 0.

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Chapter 4

Drinfeld Associator and Single-Valued MZVs

A crucial observation by Deligne is that one can attach tangential base point on P \ {0, 1, ∞} to study the torsors of paths and the motivic fundamental group. Let 10 and (−1)1 be the tangential base point at 0 (resp. at 1) pointing to the positive (resp. negative) direction. Then the de Rham version of the Betty path from 10 going straight to (−1)1 gives rise to the Drinfeld associator whose coefficients are all expressible by the MZVs. 4.1 Drinfeld Associator 4.1.1

KZ-Equations: A General Setup

For future reference, we consider the general case by setting X = {x1 , . . . , xN } to be an alphabet of N +1 letters and X∗ the set of words on X including the empty word 1. Let ChhXii be the algebra of non-commutative formal power series in X∗ . Fix an injection j : X ,→ C and set pi = j(xi ) for all i = 0, . . . , N . Consider the differential equation N X xi F (z). (4.1) F 0 (z) = z − pi i=0 TN For each i, fix a half line `(pi ) starting from pi such that i=0 `(pi ) = ∅. SN Let U = C\ i=0 `(pi ) and fix a branch of logarithm log(z −p0 ) on C\`(p0 ). The following general theorem guarantees the existence of the Drinfeld associator. Theorem 4.1.1. The differential equation (4.1) has a unique solution L(z) on U such that
L(z) = f0 (z) exp x0 log(z − p0 ) , SN where f0 (z) is a holomorphic function on C \ i=1 `(pi ) with f0 (p0 ) = 1. SN Moreover, every holomorphic solution of (4.1) on C \ i=0 {pi } is equal to L(z)C for some C ∈ ChhXii depending only on pi ’s but not on z. 97

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Proof. Set L(z) = alent to

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P

w∈X∗

Lw (z)w. Note that L(z) satisfying (4.1) is equiv∂ Lw (z) Lx w (z) = ∂z i z − pi

for all i = 0, . . . , N and all w ∈ X∗ . We now recursively define functions Lw (z) satisfying (4.1) for all non-empty words w ∈ X∗ not ending in the letter x0 . For such words, the limiting condition is just limz→p0 Lw (z) = 0. If we write w = xs01 −1 xi1 . . . xs0d −1 xid , where 1 ≤ i1 , . . . , id ≤ N, then in a neighborhood of p0 the following function satisfies (4.1): Lw (z) =

X k1 >···>kd

d (−1)d Y  z − p0 km −km+1 k1s1 · · · kdsd m=1 pim − p0

(kd+1 = 0),

which converges absolutely for |z − p0 | < min{|pi1 − p0 |, . . . , |pid − p0 |}. The functions Lw (z) can be extended analytically to the whole of U by the recursive formula Z z Lw (t) dt, ∀i = 0, . . . , N. Lxi w (z) = 0 t − pi P Now we write f0 (z) = w∈X∗ Λw (z)w where  1, if w = 1; Λw (z) = (4.2) 0, if w = xn0 , n ∈ N;

and if w = xs01 xi1 . . . xs0d xid xn0 , where s1 , . . . , sd ≥ 0 and 1 ≤ i1 , . . . , id ≤ N, then d   Y X kj Λw (z) = (−1)n Lxk1 x ...xkd x (z). (4.3) i1 id 0 0 sj j=1 sj ≤kj ,|k|=|s|+n


Then it is not hard to see that L(z) = f0 (z) exp x0 log(z − p0 ) satisfies equation (4.1). Note that f0 (z) is holomorphic in a neighborhood of p0 since its coefficients are just linear combinations of the functions Lw (z). Finally, the uniqueness can be proved in a straightforward manner. This completes the proof of the theorem. Definition 4.1.2. Let x0 and x1 be two non-commuting symbols. Let G0 and G1 be the analytic solutions of the Knizhnik–Zamolodchikov (KZ) differential equation   x0 x1 G0 (z) = + G(z), (4.4) z z−1

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that are defined for 0 < z < 1, such that

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G0 (z) ∼ z x0 near z = 0

and G1 (z) ∼ (1 − z)x1 near z = 1.

This means both G0 (z) and G1 (z) are formal series in x0 and x1 with coefficients given by analytic functions of z in (0, 1). Moreover, G0 (z)e−x0 log z (resp. G1 (z)e−x1 log(1−z) ) has an analytic continuation into a neighborhood of z = 0 (reps. z = 1) and becomes 1 at this point. The Drinfeld associator is define by X Φ(x0 , x1 ) := Φ[w]w := G−1 (4.5) 1 G0 . w∈X∗

Here and in what follows f [w] always means the coefficient of w in f . Remark 4.1.3. The existence of G0 (z) and G1 (z) follows from Thm. 4.1.1 by taking p0 = 0 and 1, respectively. Note that Φ(x0 , x1 ) should only depend on x0 and x1 . In order to understand this we may consider the analogous but much easier differential equation f 0 (z) = f (z) with two solutions f0 (z) and f1 (z) determined by the condition that f0 (0) = 1 and f1 (1) = 1. Then f0 (z) = ez and f1 (z) = ez−1 . So f1 (z)−1 f0 (z) = e which is a constant. Theorem 4.1.4. The Drinfeld associator satisfies the following duality relation: Φ(x0 , x1 )−1 = Φ(x1 , x0 ).

(4.6)

Proof. First observe that by z ↔ 1 − z we can rewrite Eq. (4.4) as   x1 x0 + H(z), H(z) = G(1 − z). H 0 (z) = z−1 z

−1 −1 Thus Φ−1 = G−1 can be obtained 0 G1 = H1 H0 which implies that Φ from Φ by the substitution x0 ↔ x1 .

Remark 4.1.5. Here is another way to understand Φ(x0 , x1 ). Since 1/z and 1/(1 − z) are both analytic in (0, 1) for any 0 < u < 1 there is a unique solution to Eq. (4.4) with G(u) = 1. Let Zuv (x0 , x1 ) = G(v) for v ∈ (0, 1). Since Zuu (x0 , x1 ) = 1 we can write X Zuv (x0 , x1 ) = fw (u, v)w, w∈X∗

where A is defined in Definition 3.2.1 and fw (u, v) isR an analytic function v in u and v. Define the iterated integral Iw (u, v) = u w(dt/t, dt/(t − 1))

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for any word w = w(x0 , x1 ). By the induction on the weight of the word we see easily that fw (u, v) = Iw (u, v). If w = xs01 −1 x1 · · · xs0d −1 x1 ∈ A0 (i.e., s1 ≥ 2, see Definition 3.2.2) then we have limε→0+ Iw (ε, 1 − ε) = (−1)d ζ(s1 , . . . , sd ) by Eq. (3.1). But if w is not admissible we have to regularize Iw (ε, 1 − ε). It is possible to compute this explicitly (see [385, Appendix]) but this would take us too far afield. Another high powered machinery to do this is to use some Hopf algebra fact whose proof will be omitted. 4.1.2

Some Hopf Algebras and Their Group-Like Elements

Let X = {x0 , x1 } and R := ChhXii be the Hopf algebra as defined in Example A.12 with concatenation product and with coproduct ∆
such that ∆
(xi ) = 1 ⊗ xi + xi ⊗ 1 for i = 0, 1. The counit 
is defined by 
(w) = 0 for all nonempty word w and 
(1) = 1. Note that R can be obtained from A by an extension of scalars to C and then by completion with respect to the weight grading. We now consider R
:= R/Rx0 which can be identified with the subalgebra of R generated by words ending with x1 . It can be thought as the completion of A1
⊗ C with respect to the weight grading. By the above we see that the Drinfeld associator is naturally associated with the shuffle regularization of the MZVs. Similarly we can treat the stuffle version using a ∗-version of Φ. Pn Let R∗ := Chhzn , n ∈ Nii be the Hopf algebra with ∆∗ (zn ) = k=0 zk ⊗ zn−k for all n ∈ N, where z0 = 1. Clearly, this is the completion of A1∗ ⊗ C with respect to the weight grading. Define the projection π : R −→ R
by forgetting the terms corresponding to the words ending with x0 . Note that if we set zn = xn−1 x0 for all n ∈ N then as concatenation algebras 0 R
= R∗ . Define Φ
= π(Φ). P Theorem 4.1.6. Let f∗ = 1 + w∈X∗ x1 f∗ [w]w ∈ R∗ . Then f∗ satisfies stuffle relation if and only if ∆∗ (f∗ ) = f∗ ⊗ f∗ , i.e., f∗ is group-like for ∆∗ . Proof. Throughout this proof, by abuse of notation, for any composition s = (s1 , . . . , sd ) ∈ Nd we also identify it with (−1)d xs01 −1 x1 . . . xs0d −1 x1 ∈ A1∗ when we write s ∈ A1∗ . We also write (r1 , . . . , r` ) ⇒ (s1 , . . . , sd )

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if ` ≥ d and (r1 , . . . , r` ) can be obtained from (s1 , . . . , sd ) by inserting some 0’s. By the definition, ∆∗ (f∗ ) = f∗ ⊗ f∗ means for all s, t ∈ A1∗ X f∗ [s]f∗ [t] = f∗ [r10 + r100 , . . . , r`0 + r`00 ]. (4.7) (r10 ,...,r`0 )⇒s,(r100 ,...,r`00 )⇒t max{dp(s),dp(t)}≤`≤dp(s)+dp(t) rj0 +rj00 >0 ∀j=1,...,`

Extending f∗ linearly to A1∗ we need to show Eq. (4.7) is equivalent to f∗ [s]f∗ [t] = f∗ [s ∗ t] for all s, t ∈ A1∗ .

(4.8)

Assuming Eq. (4.7) we now establish Eq. (4.8) by induction on |s| + |t|. If either s or t is 1 then this is obvious. In general, we have X f∗ [m, s]f∗ [n, t] = f∗ [r10 + r100 , . . . , r`0 + r`00 ]. (r10 ,...,r`0 )⇒(m,s),(r100 ,...,r`00 )⇒(n,t) max{dp(s),dp(t)}+1≤`≤dp(s)+dp(t)+2 rj0 +rj00 >0 ∀j=1,...,`

On the right-hand side we have three cases to consider: (r10 , r100 ) = (m, 0), (r10 , r100 ) = (0, n), and (r10 , r100 ) = (m, n). By the induction assumption we then get       f∗ [m, s]f∗ [n, t] = f∗ m, s ∗ (n, t) + f∗ n, (m, s) ∗ t + f∗ m + n, s ∗ t   = f∗ (m, s) ∗ (n, t) . This completes the proof that ∆∗ (f∗ ) = f∗ ⊗ f∗ implies Eq. (4.8). On the other hand, it is not too hard to check that Eq. (4.8) implies Eq. (4.7) (see Exercise 4.3) which means ∆∗ (f∗ ) = f∗ ⊗ f∗ . This finishes the proof of the theorem. Since the ∗-regularized MZVs ζ∗ (s1 , . . . , sd ; T ) satisfy stuffle relations the next result follows immediately. Corollary 4.1.7. Let ∞ X Φ∗ = 1 +

X

(−1)d ζ∗ (s1 , . . . , sd )zs1 . . . zsd .

d=1 s1 ,...,sd ∈N

Then ∆∗ (Φ∗ ) = Φ∗ ⊗ Φ∗ .

P Theorem 4.1.8. Let f = 1 + w∈x0 X∗ x1 f [w]w ∈ (Rcv , ∆
). Then f [w] satisfies the shuffle relation if and only if ∆
(f ) = f ⊗ f , i.e., f is grouplike. Similar result holds for (R
, ∆
), too. Proof. The computation is straightforward so we leave it as an exercise (see Exercise 4.4).

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We leave the proof of next two technical lemmas to the interested reader (see Exercise 4.5). Lemma 4.1.9. Let R∗T = R∗ ⊗C C[T ]. Any group-like element fcv of (Rcv , ∆∗ ) is the image of a group-like element of (R∗T , ∆∗ ) under the specialization T = 0. Moreover, two such elements f1 and f2 are related by f2 = exp((λ2 − λ1 )z1 )f1 , where λ1 and λ2 are the coefficients of z1 in f1 and f2 respectively. Similarly one can prove, T Lemma 4.1.10. Let R
= R
⊗C C[T ]. Any group-like element fcv of T , ∆
) under the (Rcv , ∆
) is the image of a group-like element of (R
specialization T = 0. Moreover, two such elements f1 and f2 are related by

f2 = exp((λ2 − λ1 )x1 )f1 , where λ1 and λ2 are the coefficients of x1 in f1 and f2 respectively. Theorem 4.1.11. The Drinfeld associator Φ(x0 , x1 ) is the only element in the Hopf algebra (R, ∆
, 
) such that (i) ∆
(Φ) = Φ ⊗ Φ ( i.e. Φ is group-like, see Definition A.6) (ii) Φ[x0 ] = Φ[x1 ] = 0 (iii) Φ[w] = (−1)d ζ(s1 , . . . , sd ) ∀w = xs01 −1 x1 · · · xs0d −1 x1 ∈ A0 . Proof. This follows from Lemma 4.1.10 easily. Remark 4.1.12. In Chap. 13 we will present a more general statement in Thm. 13.4.1. It is easy to see that the shuffle relations between the MZVs are direct consequences of the fact that Φ is group-like, namely, ∆
(Φ) = Φ ⊗ Φ. See Thm. 4.1.8 and Exercise 4.4. Using Thm. 4.1.11 we can prove the following result which provides the recursive relations one may use to compute all the coefficients of Φ. Proposition 4.1.13. We have (i) For w = xs01 −1 x1 · · · xs0d −1 x1 ∈ A0 , we have Φ[w] = (−1)d ζ(s1 , . . . , sd ).

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Drinfeld Associator and Single-Valued MZVs

n 1 m n (ii) For w = xm 1 vx0 6∈ A with m + n > 0, we may set w = x1 vx0 = m n x1 l1 · · · lp x0 where lj ∈ {x0 , x1 } and if p > 0 then l1 6= x1 , lp 6= x0 . Further set l1 · · · lp xs0 = l1 · · · lq . Then    0, if mn = p = 0;    q   1 X  Φ[xm−1 l1 · · · li x1 li+1 . . . lq ], if m > 0; − 1 (4.9) Φ[w] = m  i=1  p   1X    Φ[l1 · · · li−1 x0 li · · · lp xn−1 ], if m = 0, n > 0. 0 −n i=1

Proof. We only need to prove (ii). Since Φ is group-like one has ∆
Φ = Φ ⊗ Φ.

(4.10)

First we assume mn = p = 0. We may further assume m = 0 (n = 0 is completely similar). We now show that Φ[xn0 ] = 0 for all n ∈ N by induction on n. Suppose Φ[xn−1 ] = 0 for some n ≥ 2. By Thm. 4.1.11 we 0 have Φ[x0 ] = 0 and by Eq. (4.10) ∆
(1) + Φ[x20 ](∆
(x0 ))2 + · · ·

= 1 ⊗ 1 + Φ[x20 ](x0 ⊗ 1 + 1 ⊗ x0 )2 + · · ·

= (1 + Φ[x0 ]x0 + · · · ) ⊗ (1 + Φ[x0 ]x0 + · · · ).

We now compare the coefficient of xn−1 ⊗ x0 on the two sides. The only 0 way to produce this on the left-hand side is by Φ[xn0 ]∆
(xn0 ) which in fact produces n copies of it. On the right-hand side the only way is to use Φ[xn−1 ]xn−1 ⊗ Φ[x0 ]x0 . Hence nΦ[xn0 ] = Φ[xn−1 ]Φ[x0 ] = 0 and therefore 0 0 0 n Φ[x0 ] = 0. By similar argument if m > 0 then one can compare the coefficient of x1 ⊗ (xm−1 l1 · · · lq ) of the two sides of Eq. (4.10) (which is 0) and find the 1 relation (4.9). Finally, if m = 0 and n > 0 then one may similarly consider the coefficient of (l1 · · · lp xn−1 ) ⊗ x0 in Eq. (4.10). This finishes the proof 0 of the proposition. Theorem 4.1.14. If w = {0}s1 −1 1 · · · {0}sd −1 1{0}n−1 , then  d  X Y kj + sj − 1 n Φ[w] = (−1) ζ(s1 + k1 , . . . , sd + kd ). kj j=1 k1 +···+kd =n

Proof. This can be proved easily by induction on n using Prop. 4.1.13 so it is left to the interested reader.

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Theorem 4.1.15. We have Φ
= Λ(x1 )Φ∗ where Λ is defined by Eq. (3.17): ! ∞ X (−1)n ζ(n) n u . Λ(u) = exp n n=2 Proof. Let Rcv := R∗ /z1 R∗ ∼ = Chhzn , n ≥ 2ii which is also equal to R∗ /x1 R∗ . Let πcv : R∗ → Rcv be the projection map by forgetting all the terms corresponding to the words beginning with x1 . The generating series of ζ(s1 , . . . , sd ) is defined as follows: X Lcv := (−1)d ζ(s1 , . . . , sd )zs1 · · · zsd ∈ (Rcv , ∆∗ ), s1 ,··· ,sd ∈N,s1 ≥2

:=

X s1 ,··· ,sd ∈N,s1 ≥2

(−1)d ζ(s1 , . . . , sd )x0s1 −1 x1 · · · x0sd −1 x1 ∈ (Rcv , ∆
).

Since MZVs satisfy the stuffle relations we see that Lcv is a group-like elements in (Rcv , ∆∗ ) by a similar argument as in the proof of Thm. 4.1.6. On the other hand, since MZVs also satisfy the shuffle relations Lcv is also a group-like elements in (Rcv , ∆
) by Thm. 4.1.8. T . Clearly L is Let L := exp(T z1 )·Φ∗ ∈ R∗T and I := exp(T x1 )·Φ
∈ R
group-like for ∆∗ and I is group-like for ∆
. Moreover, it is straightforward to check the coefficients of z1 = x1 in L and I are both equal to T . By the uniqueness guaranteed by Lemma 4.1.9 and Lemma 4.1.10 X X L=1+ ζ∗ (w; T )w, I = 1 + ζ
(w; T )w w∈X∗ x1

w∈X∗ x1

since both of the right-hand sides are group-like (with respect to ∆∗ by Thm. 4.1.6 and ∆
by Thm. 4.1.8, respectively). By Thm. 3.3.21 we see that ρ(L) = I = exp(T x1 ) · Φ
. But by the definition of ρ in Lemma 3.3.17 ρ(L) = ρ(exp(T z1 ) · Φ∗ ) = Λ(x1 ) exp(T x1 )Φ∗ = exp(T x1 ) · Λ(x1 )Φ∗ . The theorem follows immediately. 4.1.3

Grothendieck–Teichm¨ uller Group

To have a better understanding of the Drinfeld associator Φ and also for future reference we need the following definitions.

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Definition 4.1.16. Let K be a field of characteristic 0. For n ≥ 2, denote by tn the completed K-Lie algebra with generators tij (i 6= j, 1 ≤ i, j ≤ n) subject to the following (infinitesimal) pure braids relations 1 : tij = tji , [tij , tik + tjk ] = 0 and [tij , tkl ] = 0

for all distinct i, j, k, l. (4.11) 0 ij Let tn be the Lie subalgebra of tn with the generators t and the same P relations as tn plus one more relation 1≤i 0 be a formal real variable (which is germain to the Planck constant in theoretical physics) and ~ = h/2πi. Let g be a Lie algebra (see Appendix B) over K[[h]] which as a K[[h]]-module is isomorphic to V[[h]] for some vector space V over K. Let t ∈ g ⊗ g be a g-invariant symmetric tensor, i.e., t = τ (t) with τ (x ⊗ y) = y ⊗ x. For any positive integer n ≥ 3, we denote by tij the image of t under the (i, j)-th embedding g ⊗ g → g⊗n . For example, if n = 3 then t12 = t ⊗ 1, t23 = 1 ⊗ t ∈ g⊗3 . Lemma 4.1.24. The tij satisfy all the pure braids relations in Eq. (4.11). Proof. This is straightforward by the symmetry of t. Definition 4.1.25. Let ~ > 0 be a real number. Let Φh = Φh (t12 , t23 ) := G−1 1 G0 , where G0 and G1 are the analytic solutions of the KZ differential equation  12  t t23 G0 (z) = ~ + G(z), (4.20) z z−1 that are defined for 0 < z < 1, such that 12

G0 (z) ∼ z ~t

near z = 0

23

and G1 (z) ∼ (1 − z)~t

near z = 1.

Here, f (z) ∼ 1 near z = 0 means that there is an expansion of f (z) = 1 + f1 (z)h + f2 (z)h2 + · · · such that fj (z) are all analytic near z = 0.

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Remark 4.1.26. By the definition Φh is an invertible element of (U g)⊗3 . From Definition 4.1.2 we see that if h = 2πi (or, equivalently, if ~ = 1) then Φ2πi (t12 , t23 ) = Φ(t12 , t23 ). The KZ system (4.20) has can be generalized to X tij ∂W =~ W, i = 1, . . . , n, ∂zi zi − zj

(4.21)

j6=i

where W (z1 , . . . , zn ) ∈ (U g)⊗n . Since ∂W/∂z1 + · · · + ∂W/∂zn = 0 the function W depends only on the difference zi − zj . Another restriction Pn P ij i=1 zi ∂W/∂zi = ~ 1=i 1 is obvious. An easy application of geometric series summation formula yields the case s1 = 1 immediately. Let ω0 (t) = dt/t and ω1 (t) = dt/(t − 1). For any path γ : [0, 1] → C\{0, 1} from y = γ(0) to z = γ(1) and a1 , . . . , an ∈ {0, 1}, we define Z Iγ (y; a1 . . . an ; z) = γ ∗ ωa1 (t1 ) ∧ . . . ∧ γ ∗ ωan (tn ) 0 M and have finite limits when z → ∞ then we define the regularized value of f (z) at ∞ to be regz=∞ f (z) = lim f0 (z). z→∞

Lemma 4.4.2. For any single-valued functions f and g defined near a ∈ C ∪ ∞, we have
regz=a f (z)g(z) = regz=a f (z) · regz=a g(z), provided all the regularized values are defined. Proof. This is clear from the definition. By the definition we see that Lw (0) := regz=0 Lw (z) = 0

∀w 6= 1.

(4.51)

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Definition 4.4.3. We set Lw (1) := regz=1 Lw (z) and call it a singlevalued MZV (SVMZV). Lemma 4.4.4. Let MZsv Z be the set of all Z-linear combinations of SVMZVs. Then MZsv is a sub-ring of MZZ , namely, Lu (1) Lv (1) ∈ MZsv Z Z ∗ for all u, v ∈ X . Proof. Without loss of generality, let u, v ∈ X∗ be two non-empty words. Then by Lemma 4.4.2 Lu (1) Lv (1) = regz=1 Lu (z) · regz=1 Lv (z)
= regz=1 Lu (z) Lv (z) = regz=1 Lu
v (z) = Lu
v (1) where we have extended L over ZhXi as a Z-linearly map. So the set MZsv Z is closed under multiplication, as desired. Example 4.4.5. An easy computation by Maple shows that up to weight sv ten MZsv Q := MZZ ⊗ Q is generated by ζ(3), ζ(5), ζ(7), ζ(9). At weight 11 we have two generators: ζ(11) and 4 6 g533 = ζ(5, 3, 3) − ζ(5)ζ(2)3 + ζ(7)ζ(2)2 + 45ζ(9)ζ(2). 7 5 More precisely, up to weight 11 the ring MZsv Z over Z is spanned by 2ζ(3), 1 1 ζ(5), 2ζ(3)2 , 81 ζ(7), ζ(3)ζ(5), 72 ζ(9), 43 ζ(3)3 , 18 ζ(3)ζ(7), 21 ζ(5)2 , 384 ζ(11), 1 3 2 2 (g + ζ(11)) + ζ(3) ζ(5), and 2ζ(3) ζ(5). 533 5 2 sv by SVMPLs. Recall Let Lssv be over h the shuffle-algebra i h MZZ spanned i 1 1 1 1 that O = Q z, z , z−1 and O = Q z¯, z¯ , z¯−1 . Define two bi-differential algebras

MZsv Z,n (Z),

Lssv n

A := O · O · Ls,

Asv := O · O · Lssv .

(4.52)

Let and Asv n denote the subspaces of total weight n. Due to the difficulty in transcendental number theory we do not know whether these subspaces for different n are Q-linearly independent. Remark 4.4.6. The only difference between A (resp. Ls) and Asv (resp. Lssv ) is the coefficients for the former are in C while those for the latter are in MZsv Z . By the definition, Asv is closed under ∂z by Thm. 4.3.2(iii). For ∂z¯, by definition (4.50) we see that   x0 y1 ∂z¯ LX (z) = LX + z¯ z¯ − 1

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121

which yields X 1 δa,0 Lw (z) + Lu (z)y1 [v] z¯ z¯ − 1 wa=uv X Lw (z) 1 = + Lu (z)(y1 [v] − δx1 ,v ), z¯ − a z¯ − 1 wa=uv

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∂z¯ Lwa (z) =

(4.53)

where y1 [v] is the coefficient of the word v in the series y1 . R Definition 4.4.7. For P ∈ A, let 0 P (z)dz be Rthe unique function F ∈ A, if it exists, such that ∂z F (z) = RP (z) with 0 denoting the regularized value regz=0 F (z) = 0. Similarly, 0 P (z)d¯ z is the unique function G(z) ∈ A, if it exists, such that ∂ G(z) = P (z). For b ∈ {1, ∞}, we define z ¯ R R P (z)dz = F (z) − F (b) and P (z)d¯ z = G(z) − G(b), where F (b) and b b G(b) are regularized values. Lemma 4.4.8. We have a well-defined integral operator on A, Z Z dz : A −→ A, F (z) 7−→ F (z)dz. 0 0 R Moreover, 0 dz maps Asv into Asv . Proof. For the first sentence, we want to show that the images of the operator lie in A. First, note that for a ∈ {0, 1} and n 6= −1 Z Z (z − a)n+1 L1 (z) dz = L1 (z − a), L1 (z)(z − a)n dz = , n+1 0 0 z−a both of which are in Asv . For general w, by Thm. 4.3.2(iii), we have Z Lw (z) dz = Law (z) ∈ Asv . 0 z−a For any n ∈ Z (n 6= −1) and w = aw0 (a = 0, 1), we have Z Z (z − a)n+1 (z − a)n+1 Lw (z)(z − a)n dz = Lw (z) − Lw0 (z) dz ∈ A n+1 n+1 0 0 R by induction on |w|. This shows that 0 dz isR a well-defined operator on A. The claim for the stability of Asv under 0 dz follows from the same argument. Proposition 4.4.9. Let S3 be the group of M¨ obius transformations persv muting {0, 1, ∞}. Then Ls is stable under S 3 . The integral operator n R dz/(z − a) (a ∈ {0, 1}, b ∈ {0, 1, ∞}) satisfies b Z sv dz/(z − a) : Lssv (4.54) n ,→ Lsn+1 . b

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Furthermore,

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∂z Lssv n =

Lssv Lssv n−1 + n−1 . z z−1

(4.55)

Proof. First, Eq. (4.55) follows Reasily from R Thm. 4.3.2(iii). The stability of z Lssv under z → z−1 and under 0 dz/z, 0 dz/(z − 1) follows from straightforward computations (see Exercises 4.27 and Lemma 4.4.8). If |w| = n and b = 1 or ∞ we have Z Z Lw (z) Lw (z) dz = dz + Law (b) ∈ Lssv n+1 , 0 z−a b z−a which implies Eq. (4.54). z and z → 1 − z we now only Since the group S3 is generated by z → z−1 sv need to prove the stability of Ls under z → 1−z. We proceed by induction on the total weight. In weight zero this is trivial. Assume P ∈ Lssv n . Then by Eq. (4.55),  Z  Q0 (z) Q1 (z) + P (z) = dz + P (1) z z−1 1

sv where Q0 , Q1 ∈ Lssv n−1 . Since Q0 (1 − z), Q1 (1 − z) ∈ Lsn−1 by the induction we finally get  Z  Q0 (1 − z) Q1 (1 − z) P (1 − z) = + dz + P (1) ∈ Lssv n z − 1 z 0

by Eq. (4.54). Proposition 4.4.10. The series y1 ∈ MZsv Z hhXii. Proof. We show that y1 [w] ∈ MZsv Z hhXii by the induction on the weight of w ∈ X∗ . Suppose w has weight n. By (4.53), for any word u, sv limz→0 z¯∂z¯ Lua (z) = δa,0 Lu (0) ∈ MZsv under Z . By the stability of Ls z → 1 − z we obtain lim (¯ z − 1)∂z¯ Lw (z) = lim z¯∂z¯ Lw (1 − z) ∈ MZsv Z .

z→1

z→0

On the other hand, from (4.53) we obtain X X lim (¯ z − 1)∂z¯ Lw (z) = Lu (1)y1 [v] = y1 [w] + Lu (1)y1 [v]. z→1

w=uv

Hence y1 ∈ MZsv Z hhXii by induction.

w=uv |u| 0 the constraint (1/i)×Eq. (4.73)= (1/j)×Eq. (4.74) i,j−1 determines recursively the coefficient ci−1,j k,l,m,n (resp. ck,l,m,n ) if i < j (resp. i > j). 4.11. Complete the proof of Lemma 4.1.28. 4.12. Complete the proof of Eq. (4.31) by mimicking the proof of the first 23 identity W2 = W3 eht /2 . 4.13. Prove Eq. (4.15) by using Eq. (4.12). 4.14. Prove that f0 (z) defined by (4.2) and (4.3) indeed provides a solution in Thm. 4.1.1. 4.15. Prove the uniqueness of the solution in Thm. 4.1.1. 4.16. Show that by the change of variables z − p0 → 1/(z − p0 ) one obtains a solution L∞ (z) to Eq. (4.1) on U which corresponds to the point at infinity. Fix any branch of log z on U . Show that L∞ (z) ∼ exp(x∞ log z) PN where x∞ = i=0 xi . 4.17. Show that for any w ∈ X∗ and a ∈ {0, 1} X 1 Lw (z) + Lu (z)(y1 [v] − δx1 ,v ). ∂z¯ Lwa (z) = z¯ − a z¯ − 1 wa=uv 4.18. For any w ∈ {0, 1}∗ (or w ∈ X∗ ), show that Z Lw (t) Law (z)(= Lxa w (z)) = dt, a ∈ {0, 1}, t−a with the initial value Lw (0) = 0 unless w = {0}n (= xn0 ) when Lw (0) = logn z n! .

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4.19. Show that F (x0 , x1 ) defined by Eq. (4.46) satisfies F ≡ 0 modulo words of depth two. Moreover, it has the following explicit expansion

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F (x0 , x1 ) = x1 + ζ(2)[[x0 , x1 ], x1 ] + ζ(3)[x1 , u3 ] + · · · where u3 is the coefficient of ζ(3) in Φ(x0 , x1 ) given by Exercise 4.2. Then show that by Eq. (4.47) y1 = x1 + 2ζ(3)[x1 , u3 ] + · · · 4.20. Show that if z1 satisfies Eq. (4.47) then it satisfies Eq. (4.45). 4.21. Show that L{0}m 1{0}n (z) =

n X

(−1)k+1

k=0

  m + k logn−k z Lim+k+1 (z). (n − k)! m

4.22. Use Exercise 4.21 to show that   n−k n X (z z¯) k+1 m + k log Lim+k+1 (z) L{0}m 1{0}n (z) = (−1) (n − k)! m k=0   m X n + k logm−k (z z¯) (−1)k+1 + Lin+k+1 (¯ z ). n (m − k)! k=0

4.23. Show that the regularizations in Definition 4.4.1 are well defined (i.e., unique). 4.24. Show that if f (z) ∈ Ls then f (z) = o(|z|− ) as z → 0, i.e., lim |z| f (z) = 0.

z→0

4.25. Show that if f (z) ∈ Ls then f (z) = o(|z| ) as z → ∞, i.e., lim |z|− f (z) = 0.

z→∞

4.26. Let P ∈ Ls and a, b ∈ {0, 1}. Show that Z  Z Z Z d¯ z d¯ z dz dz P (z) − ¯−b ¯−b 0 z −a 0 z 0 z−a 0 z  Z Z dz d¯ z P (z) − P (z) · L1 (z). = ¯−b 0 z−a 0 z z=a z=b

4.27. Prove the stability of Lssv under the transformations z → using the following identities: ←

z z−1

←

Lx0 ,x1 (1 − z) = Lx1 ,x0 (z)Φx0 ,x1 Φx0 ,y1 Ly1 ,x0 (¯ z)

= Lx1 ,x0 (z) Lx0 ,x1 (1) = Lx0 ,x1 (1) Ly1 ,y←1 (y1 ,x0 ) (z),

by

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 Lx0 ,x1

z−1 z



←

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←

= L−x0 −x1 ,x0 (z)Φx0 ,−x0 −x1 Φx0 ,−x0 −y1 L−x0 −y1 ,x0 (¯ z)

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= L−x0 −x1 ,x0 (z) Lx0 ,−x0 −x1 (1) = Lx0 ,−x0 −x1 (1) L−x0 −y1 ,y←1 (−x0 −y1 ,x0 ) (z), 



← z = Lx0 ,−x0 −x1 (z)Lx0 ,−x0 −y1 (¯ z ) = Lx0 ,−x0 −x1 (z), z−1   ← ← 1 Lx0 ,x1 = Lx1 ,−x0 −x1 (z)Φx0 ,x1 Φx0 ,y1 Ly1 ,−x0 −y1 (¯ z) 1−z = Lx1 ,−x0 −x1 (z) Lx0 ,x1 (1)

Lx0 ,x1

= Lx0 ,x1 (1) Ly1 ,y←1 (y1 ,−x0 −y1 ) (z),   ← ← 1 = L−x0 −x1 ,x1 (z)Φx0 ,−x0 −x1 Φx0 ,−x0 −y1 L−x0 −y1 ,y1 (¯ Lx0 ,x1 z) z = L−x0 −x1 ,x1 (z) Lx0 ,−x0 −x1 (1) = Lx0 ,−x0 −x1 (1) L−x0 −y1 ,y←1 (−x0 −y1 ,y1 ) (z). 4.28. Show that modulo IH the series V = Φ−1 x1 Φ satisfies V ≡ (1 − Φ00,0 + Φ00,1 + Φ10,0 − Φ10,1 )x1 (1 + Φ00,0 + Φ01,0 + Φ10,0 + Φ11,0 ). 4.29. Show that modulo IH 0 0 0 0 0 0 2V0,1 + S00,0 V1,1 ≡ 2V1,0 − V1,1 S00,0 ≡ V0,1 S00,0 + S00,0 V1,0 ≡0, 1 0 0 2V1,1 − V1,1 S10,1 + S11,0 V1,1 ≡ 0,

0 0 0 1 0 V0,1 S10,1 + S10,1 V0,1 + S00,0 (V0,1 + V1,1 ) + S10,0 V1,1 ≡ 0, 0 0 0 1 0 V1,0 S11,0 + S11,0 V1,0 + (V1,0 + V1,1 )S00,0 − V1,1 S10,0 ≡ 0, 0 0 W + (S00,0 + S10,1 )V0,0 − V0,0 (S00,0 + S11,0 ) ≡ 0,

1 0 1 1 0 where W = 2V0,0 + V0,1 S10,0 − V0,1 S00,0 + S00,0 V1,0 + S10,0 V1,0 . P a ˆ where Vˆ a is the sum of 4.30. Decompose V as V = a,b,c∈{0,1} Vˆb,c in H the subset of words of V that begins with xb , ends with xc , and contains the subword x21 exactly a times. Show that modulo IHˆ 0 ˆ0 Vˆ 0 ≡ 2Vˆ 1 − Vˆ 0 S ˆ1 ˆ1 ˆ 0 2Vˆ0,1 +S 0,0 1,1 1,1 1,1 0,1 + S1,0 V1,1 ≡ 0, 0 ˆ1 ˆ1 Vˆ 0 + S ˆ0 Vˆ 1 + S ˆ1 Vˆ 0 ≡ 0, S0,1 + S Vˆ0,1 0,1 0,1 0,0 1,1 0,0 1,1 0 ˆ1 ˆ1 Vˆ 0 ≡ 0, W − Vˆ0,0 S1,0 + S 0,1 0,0 1 0 ˆ1 ˆ0 ˆ0 ˆ 1 ˆ1 Vˆ 0 − Vˆ 1 S where W = 2Vˆ0,0 + Vˆ0,1 S0,0 + S 0,1 0,0 + S0,0 V1,0 . 0,0 1,0

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Chapter 5

Multiple Zeta Value Identities

There are plenty of known relations among the MZVs most of which have been proved rigorously in recent years with a variety of techniques. In Sec. 3.3 we have seen that the (regularized) DBSFs give rise to many linear relations over Q and conjecturally these imply all the other linear relations. In this chapter we first collect a few special families of these relations and then present some conjectural ones at the end. We will often resort to the properties of Bernoulli numbers thanks to Euler’s famous evaluation in Eq. (1.2). 5.1 Duality Relations Let k be a positive integer and s = (s1 , . . . , sd ) of weight w = |s|. Write s = (a1 + 1, {1}b1 −1 , . . . , a` + 1, {1}b` −1 ) for a1 , b1 , . . . , a` , b` ≥ 1. Then its dual is defined by s∗ = (b` + 1, {1}a` −1 , . . . , b1 + 1, {1}a1 −1 ).

(5.1)

An important relation between s and its dual is the following: dp(s) + dp(s∗ ) = |s|.

(5.2)

Example 5.1.1. One can check easily that (n + 1, {1}k−1 )∗ = (k + 1, {1}n−1 ). Recall from Definition 3.2.1 that A is the Q-algebra of words generated over x0 and x1 . In this chapter, to save space, we will write x0 = a and x1 = b. The following result provides so-called duality relation of the MZVs. 135

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Theorem 5.1.2. For every admissible composition s, we have ζ(s) = ζ(s∗ ).

(5.3)

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Using admissible words (see Definition 3.2.2), this can be expressed as ζ(w) = ζ(τ w) 0

∀w ∈ A0 ,

(5.4)

where τ is the anti-automorphism of A by exchanging a and b. Proof. By Eq. (5.1), given an admissible word w = as1 −1 b · · · asd −1 b (see Definition 3.2.1) and s∗ = t = (t1 , . . . , t` ) one checks that at1 −1 b · · · at` −1 b = absd −1 · · · abs1 −1 = τ (w) (see Exercise 5.2). Set the 1-forms a = dt/t and b = dt/(1 − t) as before. By Chen’s formula on the reversal of the iterated integrals Lemma 2.1.2(iii) followed by the substitution t → 1 − t one gets (notice d(1 − t) = −dt): Z 1 Z 0 ζ(s) = (−1)|s| w = basd −1 · · · bas1 −1 0

1

= (−1)|s|

Z 0

1

absd −1 b · · · abs1 −1 = ζ(s∗ ),

as desired. 5.2 Cyclic Sum Relation First we define a cyclic equivalence classes of multiple indices on the set I(w, d) = {(k1 , . . . , kd )|k1 + · · · + kd = w, k1 , . . . , kd ≥ 1}. Definition 5.2.1. We say two elements of I(w, d) are cyclically equivalent if they are cyclic permutations of each other, i.e., for the permutation σ = (1, . . . , d) and j = 1, . . . , d, we define (k1 , . . . , kd ) ≡ (σ j (k1 ), . . . , σ j (kd )) whose equivalence class is denoted by [(k1 , . . . , kd )]. Let C(w, d) be the set of cyclic equivalence classes of I(w, d). The main idea of the proof of the cyclic sum relation is to use partial fractions. Already applied by Euler this technique is ubiquitous in a lot of proofs of the MZV identities. We first need a lemma. Let N0 = N ∪ {0} be the set of nonnegative integers. For k1 , . . . , kd ∈ N and kd+1 ∈ N0 , let X 1 A(k1 , . . . , kd ) = (n1 − nd+1 )nk11 . . . nkdd n1 >n2 >···>nd >nd+1 ≥0 and B(k1 , . . . , kd , kd+1 ) =

X

1 k

n1 >n2 >···>nd >nd+1 >0

d+1 (n1 − nd+1 )nk11 . . . nkdd nd+1

.

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Lemma 5.2.2. The infinite sum A(k1 , . . . , kd ) converges when one of k1 , . . . , kd is greater than 1, while B(k1 , . . . , kd , kd+1 ) converges when one of k1 , . . . , kd , kd+1 + 1 is greater than 1. Proof. By the definition B(k1 , . . . , kd , 0) = A(k1 , . . . , kd ) − ζ(k1 + 1, k2 , . . . , kd ).

Applying the partial fraction identity   1 1 1 1 = − n1 (n1 − nd+1 ) nd+1 n1 − nd+1 n1

(5.5)

(5.6)

to B(k1 , . . . , kd , kd+1 ) we get

B(k1 , . . . , kd , kd+1 ) = B(k1 −1, k2 , . . . , kd , kd+1 +1)−ζ(k1 , . . . , kd , kd+1 +1). (5.7) Applying Eq. (5.6) to B(1, k2 , . . . , kd , kd+1 ) we get   X 1 1 1 − kd kd+1 +1 k2 n1 − nd+1 n1 n1 >n2 >···>nd >nd+1 >0 n2 . . . nd nd+1   ∞ X X 1 1 1 = − kd kd+1 +1 k2 n1 − nd+1 n1 n1 =n2 +1 n2 >···>nd >nd+1 >0 n2 . . . nd nd+1 =

k2 n2 >···>nd >nd+1 >0 n2

X

kd+1 +1 . . . nkdd nd+1

j=0 k

n2 >···>nd >nd+1 >j≥0

1 n2 − j

1

X

=

nd+1 −1

1

X

d+1 (n2 − j)nk22 . . . nkdd nd+1

+1

and so B(1, k2 , . . . , kd , kd+1 ) = A(k2 , . . . , kd , kd+1 + 1).

(5.8)

By Eq. (5.5) B(k1 , . . . , kd , kd+1 ) ≤ B(k1 , . . . , kd , 0) ≤ A(k1 , . . . , kd ).

If k1 ≥ 2 then B(k1 , . . . , kd+1 ) converges whenever A(k1 , . . . , kd ) does; otherwise, if k1 = 1 then Eq. (5.8) yields B(k1 , . . . , kd+1 ) = A(k2 , . . . , kd , kd+1 + 1). So the statement about the B’s follows from the one about the A’s. Turning to the first assertion of lemma we clearly only need to consider the case k1 + · · · + kd = d + 1 when A(k1 , . . . , kd ) are among the largest. Now X 1 A(2, 1, . . . , 1) = 2 n1 (n1 − nd+1 )n2 . . . nd n1 >n2 >···>nd+1 ≥0

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≤

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1 . . . nd

X

n2 jn2 n1 >n2 >···>nd >0 1 n1 ≥j>0

= ζ(3, {1}d−1 ) + dζ(2, {1}d ) +

d−1 X i=1

ζ(2, {1}i−1 , 2, {1}d−i−1 )

by inserting j into all possible places inside (0, n1 + 1). So A(2, 1, . . . , 1) converges and by Eqs. (5.5) and (5.8) we have A(1, 2, 1, . . . , 1) = B(1, 2, 1, . . . , 1, 0) + ζ(2, 2, 1, . . . , 1) = A(2, 1, . . . , 1, 1) + ζ(2, 2, 1, . . . , 1). We can repeat this idea and show that all the sums A(1, . . . , 1, 2, 1, . . . , 1) converge. This completes the proof of the lemma. The next result provides the cyclic sum relations of the MZVs. Theorem 5.2.3. For any k = (k1 , . . . , kd ) ∈ Nd with some ki ≥ 2, d X

ζ(kj + 1, kj+1 , . . . , kd , k1 , . . . , kj−1 )

j=1 kj −2

X

=

X

{j:kj ≥2} q=0

ζ(kj − q, kj+1 , . . . , kd , k1 , . . . , kj−1 , q + 1).

Equivalently, X ζ(p1 + 1, p2 , . . . , pd ) = (p1 ,...,pd )∈[k]

X

(5.9)

ζ(q1 + 1, q2 , . . . , q|k|−d ).

(q1 ,...,q|k|−d )∈[k∗ ]

(5.10) Proof. We will prove Eq. (5.9) and leave the interested reader to check that Eq. (5.9) and Eq. (5.10) are equivalent (see Exercise 5.4). Let us first show that A(k1 , . . . , kd ) − A(k2 , . . . , kd , k1 ) = ζ(k1 + 1, k2 , . . . , kd ) −

kX 1 −2 j=0

ζ(k1 − j, k2 , . . . , kd , j + 1)

(5.11)

where the sum on the right is understood as 0 if k1 = 1. We have A(k1 , . . . , kd ) − ζ(k1 + 1, k2 , . . . , kd )

=B(k1 , . . . , kd , 0)

(by Eq. (5.5))

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=B(k1 − 1, k2 , . . . , kd , 1) − ζ(k1 , . . . , kd , 1) = · · ·

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=B(1, k2 , . . . , kd , k1 − 1) − =A(k2 , . . . , kd , k1 ) −

kX 1 −2 j=0

kX 1 −2 j=0

mzv-mpl

(by Eq. (5.7))

ζ(k1 − j, k2 , . . . , kd , j + 1) (by Eq. (5.7))

ζ(k1 − j, k2 , . . . , kd , j + 1)

(by Eq. (5.8)),

which yields Eq. (5.11). Equation (5.9) now follows from adding Eq. (5.11) over all cyclic permutations of the sequence (k1 , . . . , kd ). This completes the proof of the theorem. An interesting corollary of the cyclic sum relations is so-called sum relations: Corollary 5.2.4. For any positive integers w and d, we have X ζ(k1 + 1, k2 , . . . , kd ) = ζ(w + 1).

(5.12)

k1 +···+kd =w

Proof. Indeed, if d = 1 then Eq. (5.12) is trivial. Suppose it is true when the depth on the left-hand side is d. Applying cyclic permutation of the indices (k1 , . . . , kd ) to the left-hand side of Eq. (5.12), we get by Thm. 5.2.3 d X X

dζ(w + 1) =

ζ(kj + 1, kj+1 , . . . , kd , k1 , . . . , kj−1 )

|k|=w j=1 kj −2

X

=

X

X

|k|=w {j|kj ≥2} q=0

=d

1 −1 X kX

|k|=w q=1

=d

X

ζ(kj − q, kj+1 , . . . , kd , k1 , . . . , kj−1 , q + 1)

ζ(k1 − q + 1, k2 , . . . , kd , q)

ζ(k1 + 1, k2 , . . . , kd , q).

|k|+q=w

This proves the sum relation Eq. (5.12) by the induction. With the notation of the Hoffman algebras as in Sec. 3.2, we define, for k ≥ 1, A1 (k) := hw : word w ∈ {a, b}∗ ab, |w| = kiQ .

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So an element in A1 (k) is a homogeneous polynomial of degree k with each monomial given by a word ending with b but not a b-power. Define a S Q-linear map ρ0 : k≥1 A1 (k) → A0 by setting


ρ0 zk1 · · · zk` =

j −1 ` kX X

j=1 i=1

` X zkj −i+1 zkj+1 · · · zkj+`−1 zi− zkj +1 zkj+1 · · · zkj+`−1 , j=1

where zm = zn if m ≡ n mod `. Define recursively
ρn+1 (w) = ρn (a + b)w ∀n ≥ 0. R1 Recall we have the map ζ : A0 → R defined by ζ(w) = 0 w (see Eq. (3.1)). Theorem 5.2.5. If n ∈ N0 and k ∈ N, then

{0} = ρk−1 (A11 ) ⊂ ρk−2 (A12 ) ⊂ · · · ⊂ ρ0 (A1k ) ⊂ ker ζ ∩ A0k+1 ,

where A0k+1 is the homogeneous weight k + 1 piece of A0 . Moreover, if k > n then   k 1X ϕ (2m − Lm,n ) − 2, dim ρn (A1k−n ) = k m m|k

where ϕ is Euler’s totient function and  m 2 − 1, Lm,n = Lm−1,n + · · · + Lm−n,n ,

if m = 1, . . . , n; if m ≥ n + 1.

We omit the proof of this theorem here. See [494, Cor. 12 and Thm. 15]. 5.3 Euler’s Decomposition Formula Revisited The following result is just Euler’s decomposition formula, presented already in Thm.3.1.1 whose proof uses shuffle product of the iterated integrals. But here we present another proof using partial fractions. We can also regard it as a kind of weighted sum relations of the double zeta values. Theorem 5.3.1. For all positive integers 2 ≤ s ≤ w − 2, we have w−1 X k − 1  k − 1  + ζ(k, w − k) = ζ(s)ζ(w − s). s−1 w−s−1 k=2

Proof. We start with an easy but crucial observation: for any positive integers m and n 1 m+n 1 1 = = + . (5.13) mn (m + n)mn (m + n)m (m + n)n

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141

Repeating this procedure one quickly finds that for any positive integers s and t     X k−1 1 k−1 1 1 = + . s − 1 (m + n)k nj t − 1 (m + n)k mj ms nt j,k>0, j+k=s+t

(5.14) This follows from the induction on w since     X 1 k−1 1 1 k−1 1 = + s+1 t k j t − 1 (m + n)k mj m n m s − 1 (m + n) n j,k>0, j+k=s+t     X k−1 1 k−1 1 . = + k j s − 1 (m + n) n m t − 1 (m + n)k mj+1 j,k>0, j+k=s+t

Thus we can finish the induction by the easily checked identity j

X 1 1 1 = + mnj (m + n)j m i=1 (m + n)j−i+1 ni using induction on j and Eq. (5.13). Now the theorem follows from taking P∞ the sum m,n=1 on both sides of Eq. (5.14). 5.4 Weighted Sum Relations The next result is a kind of weighted form of the sum relations of the MZVs considered in Cor. 5.2.4. We shall prove it with some generating series of the MZVs. Theorem 5.4.1. For all integer w ≥ 3, w−1 X k=2

2k ζ(k, w − k) = (w + 1)ζ(w).

(5.15)

Proof. By the stuffle relation and Thm. 5.3.1 the product ζ(s)ζ(r) can be expressed in two different ways: r+s−1 X k − 1 k − 1 ζ(r, s) + ζ(s, r) + ζ(r + s) = + ζ(k, r + s − k). r−1 s−1 k=2 (5.16) For any positive integer d, we define the generating function of the MZVs by X Gd (x1 , . . . , xd ) = x1s1 −1 · · · xdsd −1 ζ(s1 , . . . , sd ). (5.17) s1 ,...,sd ∈N,s1 >1

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Then by multiplying xa−1 y b−1 on Eq. (5.16) and then summing over a and b we get G1 (x) − G1 (y) G2 (x + y, x) + G2 (x + y, y) = G2 (x, y) + G2 (y, x) + . (5.18) x−y Replacing (x, y) by (tx, ty) and considering the coefficients of tw−2 we get w−1 X k=2

=

(x + y)k−1 (xw−k−1 + y w−k−1 )ζ(k, w − k) w−1 X

(x

k−1 w−k−1

y

+y

k−1 w−k−1

x

k=2

)ζ(k, w − k) + ζ(w)

w−2 X

xi y w−2−i .

i=0

Taking x = y = 1 and using the sum relation Cor. 5.2.4 we arrive at Eq. (5.15). There are some other families of weighted sum relations implicitly related to the above. The results in the next theorem are called restricted sum formulas. But they can be regarded as weighted sum formulas with weight factors given by either 0 or 1. Theorem 5.4.2. For all n ≥ 2, we have n−1 X 3 ζ(2k, 2n − 2k) = ζ(2n), 4

(5.19)

k=1

n−1 X k=1

1 ζ(2k + 1, 2n − 2k − 1) = ζ(2n). 4

(5.20)

Proof. Clearly Eq. (5.20) follows from Eq. (5.19) and the sum relation Eq. (5.12) which says in our case 2n−1 X ζ(k, 2n − k) = ζ(2n). k=2

Now we prove Eq. (5.19) using the following well-known formula for Bernoulli numbers (see for e.g., [143, p. 119]): n−1 X 2n (2n + 1)B2n = − B2k B2n−2k , ∀n ≥ 2. (5.21) 2k k=1

By Euler’s famous evaluation Eq. (1.2) we get  n−1 n−1  X (2πi)2n X 2n ζ(2k)ζ(2n − 2k) = B2k B2n−2k 4(2n)! 2k k=1

k=1

2

=

(2πi) n 2n + 1 (2n + 1)B2n = ζ(2n). 4(2n)! 2

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Now adding the following stuffle relations

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ζ(2k)ζ(2n − 2k) = ζ(2k, 2n − 2k) + ζ(2n − 2k, 2k) + ζ(2n)

(5.22)

for k = 1 up to n − 1 we get 2

n−1 X k=1

ζ(2k, 2n − 2k) + (n − 1)ζ(2n) =

2n + 1 ζ(2n), 2

which yields Eq. (5.19) immediately. This finishes the proof of the theorem. Theorem 5.4.3. For all positive integer n ≥ 2, we have   n−1 X 4 4n ζ(2n). (4k + 4n−k )ζ(2k, 2n − 2k) = n + + 3 6

(5.23)

k=1

For all positive integer n ≥ 2, we have n−2 X k=2

(2k − 1)(2n − 2k − 1)ζ(2k, 2n − 2k) =

3 (n − 3)ζ(2n). 4

(5.24)

Proof. We have the key identity of Bernoulli polynomials (see [143, p. 119]): n   X n Bi (x)Bn−i (y) = n(x+y −1)Bn−1 (x+y)−(n−1)Bn (x+y) (5.25) i i=0 which can be proved by observing that (setting z = x + y)  zt  text teyt t2 ezt (z − 1)t2 ezt e 2 d = = − t . et − 1 et − 1 (et − 1)2 et − 1 dt et − 1

Taking x = 1/2 and y = 0 and using the fact that Bn (1/2) = −(1 − 21−n )Bn , we get n−1 X 2n (21−2k − 1)B2k B2n−2k = 2n(1 − 21−2n )B2n − B2n . 2k k=1

By Eq. (5.21) this reduces to n−1 X 2n 21−2k B2k B2n−2k = −(2 + 41−n n)B2n . 2k k=1

Rewriting this using zeta values according to Eq. (1.2), we get n−1 X k=1

41−k ζ(2k)ζ(2n − 2k) = (2 + 41−n n)ζ(2n).

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By the stuffle relation of Eq. (5.22), this quickly leads to Eq. (5.23). P∞ Let f (t) = 1/(et − 1) = j=0 Bj tj−1 /j!. Then ∞

X tj−2 −et = (j − 1)B f (t) = t j (e − 1)2 j! j=0

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0

has only even powers of t. We have ∞

f 00 (t) = − f 0 (t) +

X 2et tj−3 = (j − 1)(j − 2)B , j (et − 1)3 j! j=0 ∞

f 000 (t) =f 0 (t) −

X 6e2t tj−4 = (j − 1)(j − 2)(j − 3)Bj . t 4 (e − 1) j! j=0

Therefore t4 f 0 (t)


2

=


t4 0 t4 e2t = f (t) − f 000 (t) . 4 − 1) 6

(et

Thus  2  ∞ ∞  j X X tj+2 tj  (j − 1)Bj t  = 1 (j − 1)Bj − (j − 1)(j − 2)(j − 3)Bj . j! 6 j=0 j! j! j=0 Comparing the coefficients of tk for even k ≥ 4, we get    X k 1 (i−1)(j−1)Bi Bj = (k−3)(k−1)kBk−2 −(k−1)(k−2)(k−3)Bk . i 6

i+j=k

Magically, the Bk−2 -terms cancel each other on the two sides since B2 = 1/6. So we have   X k 6 (i − 1)(j − 1)Bi Bj = −(k − 1)(k 2 − 5k − 6)Bk . (5.26) i i,j≥4,i+j=k

Setting k = 2n, multiplying both sides by (2πi)2n /4 and using Eq. (1.2), we get 6

n−2 X k=2

(2k − 1)(2n − 2k − 1)ζ(2k)ζ(2n − 2k) = (n − 3)(4n2 − 1)ζ(2n). (5.27)

By the stuffle relation Eq. (5.22) and the fact n−2 X k=2

(2k − 1)(2n − 2k − 1) =

(n − 3)(2n2 − 5) , 3

we finally arrive at Eq. (5.24) and conclude the proof of the theorem.

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Theorem 5.4.4. For all positive integer r and n ≥ 2r, we have

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n−r−1 X

(2n − 2r − 2)! B2k B2n−2k (2k − r − 1)!(2n − 2k − r − 1)! 2k 2n − 2k k=r+1   (r!)2 2(−1)r r!(2n − 2r − 2)! B2n + =− (2n − r − 1)! (2r + 1)! 2n ( ! !) r X r 2n − 2r − 2 B2k B2n−2k r − . 2 (−1) + 2k 2n − 2k (5.28) 2k − r − 1 2k − r − 1 k=dr/2e+1

Proof. Define a zeta function ∞ ∞ n−1 X X X Zr (s) = αr β r (α + β)−s = β r (n − β)r n−s n=1 β=1

α,β=1

by substitution n = α + β. Let Bj (x) be the jth Bernoulli polynomial. Then we have ∞ n−1 r  X X X r (−1)j nr−j−s β r+j Zr (s) = j n=1 j=0 β=1 ∞ r  X X r Br+j+1 (n) − Br+j+1 (−1)j nr−j−s = r+j+1 j n=1 j=0 =

∞ r  X X r j=0

=

j

j r−j−s

(−1) n

n=1

k=0

 r+j  r  X X r+j+1 r j=0

j

 r+j  X r+j+1

k

k=0

(−1)j

k

Bk nr+j+1−k r+j+1

Bk ζ(s + k − 2r − 1). r+j+1

Now change the order of summations and break it into three parts: (i) P0 Pr Pr Pr Pr Pr+j P2r Pr k=0 j=0 , (ii) k=1 j=0 , and (iii) j=1 k=r+1 = k=r+1 j=k−r . (i) Define r   X r xr+j+1 . f (x) = (−1)j r+j+1 j j=0 Then f 0 (x) =

r   X r j=0

j

(−1)j xr+j = xr (1 − x)r .

R1 Thus using beta function B(a, b) = 0 xa−1 (1 − x)b−1 dx = Γ(a)Γ(b)/Γ(a + b) we have Z 1 r   X r xr+j+1 (r!)2 (−1)j = xr (1 − x)r dx = B(r + 1, r + 1) = . j r+j+1 (2r + 1)! 0 j=0

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(ii) We want to show this sum is 0. Since r+j+1 /(r + j + 1) is a k polynomial of j of degree k − 1 for all k ≥ 1 it suffices to prove that for all nonnegative integer d < r we have g(1) = 0 where d X d     r r   X d r d d r j j j j g(x) = (1 − x)r . j (−1) x = x (−1) x = x dx dx j j j=0 j=0 Since r > d by Leibnitz rule it is now obvious that g(1) = 0. (iii) For each fixed integer k ∈ [r + 1, 2r], we claim that      r X r r+j r (−1)j = (−1)r (1 + (−1)k ) . j k−1 k−r−1

(5.29)

j=k−r

To show this we set h(x) :=

2r X

r X

(−1)j

k=r+1 j=k−r

      r+j r r r+j k r r+j k X X (−1)j x . x = j k−1 j k−1 j=1 k=r+1

For a polynomial f (x), we denote by Truncd f (x) the truncation of f (x) by throwing away all the terms of degree no greater than d. Then we see that  X  r+j  r X r r+j k h(x) = Truncr (−1)j x j k−1 j=1 k=1

  r+j−1 X r + j  j r = Truncr (−1) xk+1 j k j=0 k=0   r  X r x(x + 1)r+j − xr+j+1 = Truncr (−1)j j j=0 n o = Truncr x(x + 1)r (−x)r − xr+1 (1 − x)r n o = (−1)r xr+1 (x + 1)r − (x − 1)r . r X

Hence Eq. (5.29) follows immediately. Putting (i), (ii) and (iii) together we get (r!)2 ζ(s − 2r − 1) (2r + 1)!   2r X r Bk r k ζ(s + k − 2r − 1). (5.30) + (−1) (1 + (−1) ) k−r−1 k

Zr (s) =

k=r+1

On the other hand Eie [190] showed that functions like Zr (s) have analytic continuation over the whole complex plane and further they are defined

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Multiple Zeta Value Identities

at negative integers (see [189, 190]). In fact the expressions in (i) to (iii) shows clearly that this can be done using the Riemann zeta functions. More importantly, Eie showed that these special values at negative integers can also be computed using some integrals over clearly specified simplices. In our situation his theory implies that Z −x  2 r r 2n−2r−2 1 r r 2n−2r−2 Z(2r+2−2n) = J (x y (x+y) )+2J x y (x+y) dy 0

by symmetry between x and y, where for positive integers aj (j = 1, . . . , m) J m (xa1 1 . . . xamm ) =

m Y

(−1)aj

j=1

Baj +1 . aj + 1

(5.31)

For the integral, we may use substitution y = −xt to get Z −x Z 1 xr y r (x + y)2n−2r−2 =(−1)r+1 x2n−1 tr (1 − t)2n−2r−2 dt 0

0

r!(2n − 2r − 2)! . =(−1)r+1 x2n−1 (2n − r − 1)! Applying operator J, we get Z(2r + 2 − 2n) =

X a+b=2n−2r−2 r

+

(2n − 2r − 2)! Ba+r+1 Bb+r+1 a!b! a+r+1b+r+1

2(−1) r!(2n − 2r − 2)! B2n . (2n − r − 1)! 2n

Comparing with evaluation of Eq. (5.30) at s = 2r + 2 − 2n we can now finish the proof of the theorem by noticing finally that in Eq. (5.28) if r is odd and k = (r + 1)/2 then the term happens to be zero. Thus we can improve the lower limit of k from d(r + 1)/2e to dr/2e + 1. Remark 5.4.5. In [7] Agoh and Dilcher gave a formula for (Bk + Bm )n (with the classical umbral calculus notation) which leads to very similar results when k = m. Example 5.4.6. Taking r = 3 in Thm. 5.4.4, we get, for all n ≥ 2r n−4 X k=4

(2n − 8)! B2k B2n−2k (2k − 4)!(2n − 2k − 4)! 2k 2n − 2k =−

(n − 6)(2n + 1)(2n2 − 11n + 35) B2n 2n − 11 − B6 B2n−6 . 140(n − 2)(n − 3)(2n − 5)(2n − 7) 2n 6

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Theorem 5.4.7. For all positive integer r and n ≥ 2r, we have n−r−1 X

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k=r+1

ζ(2k)ζ(2n − 2k)

 = (−1)r r! +

−

r X

×

α+1

2r+1 Y

2

(r!) 2(2r + 1)! r Y

 (2n − β) ζ(2n)

β=r+1

(2k − α)

k=dr/2e+1 α=1



r n o Y (2k − α)(2n − 2k − α)

2r+1 Y

r Y

(2n − α)

α=1

(2n − β)

β=2k+1

2(2n − 2r − 2)! 2(−1)r r! + (2r − 2k + 1)! (2n − 2k − r − 1)!

 ζ(2k)ζ(2n − 2k).

(5.32)

Proof. This follows from Thm. 5.4.4 and Eq. (1.2) easily. Example 5.4.8. When r = 1 we recover the formula Eq. (5.27). When r = 2 we find for all n ≥ 4 X (2j − 1)(2j − 2)(2k − 1)(2k − 2)ζ(2j)ζ(2k) j,k≥3,j+k=n

=

1 (n − 1)(4n2 − 1)(2n2 − 13n + 30)ζ(2n) 15 − 24(n − 2)(2n − 5)ζ(4)ζ(2n − 4).

When r = 3 for all n ≥ 6 X (2j − 1)(2j − 2)(2j − 3)(2k − 1)(2k − 2)(2k − 3)ζ(2j)ζ(2k) j,k≥4,j+k=n

=

1 (n − 6)(2n − 3)(n − 1)(4n2 − 1)(2n2 − 11n + 35)ζ(2n) 35 − 240(2n − 11)(n − 3)(2n − 7)ζ(6)ζ(2n − 6).

When r = 4 for all n ≥ 8 X j,k≥5,j+k=n

ζ(2j)ζ(2k)

4 n o Y (2j − α)(2k − α)

α=1

  8 n−1 = (2n − 3)(4n2 − 1)(4n4 − 72n3 + 521n2 − 1923n + 3780)ζ(2n) 315 2 − 960(n − 3)(n − 4)(2n − 7)(2n − 9)ζ(6)ζ(2n − 6)

− 6720(n − 4)(2n − 9)(2n2 − 25n + 81)ζ(8)ζ(2n − 8).

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149

Corollary 5.4.9. For all n ≥ 4, we have n−3 2 n o X Y ζ(2k, 2n − 2k) (2n − 2k − α)(2k − α) α=1

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k=3

1 = (57n2 − 279n + 366)ζ(2n) − 12(n − 2)(2n − 5)ζ(4)ζ(2n − 4). (5.33) 2 For all n ≥ 6, n−4 3 n o X Y ζ(2k, 2n − 2k) (2n − 2k − α)(2k − α) α=1

k=4

1 = (1005n3 − 12222n2 + 48243n − 62946)ζ(2n) 2 − 120(2n − 11)(n − 3)(2n − 7)ζ(6)ζ(2n − 6).

(5.34)

For all n ≥ 8, 4 n n−5 o Y X ζ(2k, 2n − 2k) (2n − 2k − α)(2k − α) α=1

k=5

1 = (31116n4 − 631800n3 + 4846020n2 − 16543800n + 21168864)ζ(2n) 2 − 480(n − 3)(n − 4)(2n − 7)(2n − 9)ζ(6)ζ(2n − 6) (5.35) − 3360(n − 4)(2n − 9)(2n2 − 25n + 81)ζ(8)ζ(2n − 8).

Corollary 5.4.10. For all n ≥ 2, we have n−1 X n 2n − 3 k(n − k)ζ(2k, 2n − 2k) = ζ(2n) + ζ(2)ζ(2n − 2). 16 4

(5.36)

k=1

Proof. Let S be the left-hand side of Eq. (5.36). By expanding (2k−1)(2n− 2k − 1) in Eq. (5.24) we see that n−1 X (4k(n − k) − 2n + 1)ζ(2k, 2n − 2k) k=1

3 (n − 3)ζ(2n) + (2n − 3)(ζ(2, 2n − 2) + ζ(2n − 2, 2)) 4 3 − 5n = ζ(2n) + (2n − 3)ζ(2)ζ(2n − 2). 4 By Eq. (5.19) we have n−1 X 3 − 5n 4S = (2n − 1) ζ(2k, 2n − 2k) + ζ(2n) + (2n − 3)ζ(2)ζ(2n − 2) 4 k=1 n = ζ(2n) + (2n − 3)ζ(2)ζ(2n − 2) 4 which yields the corollary at once. =

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Similarly we can replace the factor k(n − k) in Cor. 5.4.10 by k r (n − k)r for any positive integer r. For example, we have Corollary 5.4.11. We have for all n ≥ 2 X

k 2 j 2 ζ(2k, 2j) =

j+k=n,j,k>0

(2n − 3)(6n − 5) n ζ(2n) + ζ(2)ζ(2n − 2) 64 16

3(2n − 5) ζ(4)ζ(2n − 4), 4 45 k 3 j 3 ζ(2k, 2j) = + (2n − 7)ζ(6)ζ(2n − 6) 8 −

X j+k=n,j,k>0

1 1 n(2n2 − 3)ζ(2n) + (2n − 3)(28n2 − 48n + 21)ζ(2)ζ(2n − 2) 256 64 3 + (2n − 5)(2n2 − 25n + 35)ζ(4)ζ(2n − 4), 16 X 315 k 4 j 4 ζ(2k, 2j) = − (2n − 9))ζ(8)ζ(2n − 8) 4

−

j+k=n,j,k>0

1 n(16n2 − 17) ζ(2n) + (2n − 3)(6n − 5)(20n2 − 36n + 17)ζ(2)ζ(2n − 2) 1024 256 3 + (2n − 5)(40n3 − 420n2 + 1050n − 777)ζ(4)ζ(2n − 4) 64 15 − (2n − 7)(8n2 − 98n + 189)ζ(6)ζ(2n − 6). 16 −

Proof. Suppose j + k = n. Then we easily see that 2 n o Y (2j − α)(2k − α) = 16j 2 k 2 + 4(5 − 6n)jk + 4(2n − 1)(n − 1).

α=1

The corollary follows from Eq. (5.19), Eq. (5.33), and Eq. (5.36) immediately. Equation (5.21) can be generalized as follows. Theorem 5.4.12. Let N ∈ N and define the unsigned Stirling numbers of the first kind by N X k=1

c(N, k)xk = x(x + 1) · · · (x + N − 1).

(5.37)

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Then for any positive integer we have  m≥N  X m Bj1 · · · BjN j1 , · · · , jN j +···+j =m 1

N

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j1 ,...,jN ≥0

= (−1)N −1 N

 X N m Bm+k−N . c(N, k) m+k−N N k=1

P

n

Proof. For any function n≥0 an t of t, we define its associated zeta funcP tion by n≥0 an /ns . Then the associated zeta function of F (t) := (1−t)−N is ∞ ∞ X X ZF (s) = ··· (n1 + · · · + nN )−s . n1 =0

nN =0

By Eie and Lai’s result [192, Prop. 5] we see that   X m ZF (−m) = (−1)N Bj 1 · · · Bj N . j1 , · · · , jN

(5.38)

j1 +···+jN =m+N j1 ,...,jN ≥0

On the other hand, by the definition  ∞  N ∞ X X X −N tn 1 n = c(N, k)nk−1 . (−t) = F (t) = (1 − t)N (N − 1)! n n=0 n=0 k=1

The associated zeta function is ∞ N X X 1 c(N, k)nk−1−s . ZF (s) = (N − 1)! n=0 k=1

Thus

ZF (−m) =

N X c(N, k) −Bm+k . (N − 1)! m + k

(5.39)

k=1

Changing the index m to m − N we see that the theorem follows from Eqs. (5.38) and (5.39) immediately.
Qk For example, putting ba1 ,...,ak = j=1 B2aj /(2aj )! , we have     X 2n + 2 1 B2n−2 , (5.40) (2n)!ba,b,c = B2n + n n − 2 2 a,b,c≥1, a+b+c=2n

X a,b,c,d≥1, a+b+c+d=2n

(2n)!ba,b,c,d

  2n + 3 4 =− B2n + n2 (2n − 1)B2n−2 . 3 3

(5.41)

These clearly yield identities between the Riemann zeta values at positive even integers. It also plays a role in the next general restricted sum relations.

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Theorem 5.4.13. Set ζ(0) = −1/2. Then for all positive integer d ≤ n, we have X ζ(2j1 , . . . , 2jd ) j1 ,...,jd >0 j1 +···+jd =n b d−1 2 c

=−

X j=0

  1 2d − 2j − 1 ζ(2j)ζ(2n − 2j). 22d−3 (2j + 1)B2j d

Proof. See [295] for details. 5.5 Derivation Relations Let a = x0 , b = x1 , and A = Qha, bi be the noncommutative polynomial algebra as in Definition 3.2.1, regarded as a graded Q-algebra with a and b both of degree one. For any word w of A, denote by |w| its total degree and by dp(w) the depth of w which is the number of occurrences of b in w. Some authors use length here, but it has an established meaning in the combinatorics of words (which is called the weight in this book). We call |w| − dp(w) the codepth of w, which gives the number of occurrences of a in w. As in Definition 3.2.2 we let A1 = Q1 ⊕ Ab and A0 = Q1 ⊕ aAb. Denote b b 1 and A b 0 the completions of the corresponding objects with respect by A, A b exchanging a and b. to the weight. Let τ be the anti-automorphism of A Applied to words, τ preserves weight and exchanges depth and codepth: b 0 (but not A b 1 ) is closed under τ . It is easy to check that note that A τ (zs ) = zs∗ where zs = zs1 · · · zsd for s = (s1 , . . . , sd ). b we mean a map δ : A b →A b (of graded As usual, by a derivation of A b It is Q-vector spaces) such that δ(uv) = δ(u)v + uδ(v) for all u, v ∈ A. straightforward to see that the commutator of two derivations is a derivab is a Lie algebra graded by the degree, tion, so the set of derivations of A b denoted by Der(A). Lemma 5.5.1. If δ is a derivation, then δ¯ := τ δτ is also a derivation (of the same degree). Proof. Let s, t ∈ Nd , and u = zs ai , v = zt aj ∈ A. Then 
 τ δτ (uv) =τ δ τ zs ai zt aj 



 =τ δ τ aj τ zt τ ai τ zs

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  =τ δ bj zt∗ bi zs∗   

 =τ δ bj zt∗ bi zs∗ + τ bj zt∗ δ bi zs∗  

 =zs ai τ δτ zt aj + τ δτ zs ai zt aj =uτ δτ (v) + τ δτ (u)v, as desired. Lemma 5.5.2. There is an isomorphism b −→ Aut(A) b exp : Der(A) ∞ X δn , δ 7−→ n! n=0 whose inverse is given by b −→ Der(A) b log : Aut(A) ∞ X (1 − σ)n σ 7−→ − . n n=1 Proof. The proof is straightforward. b To state the next theorem we now define d to be the derivation on A such that d(w) = b


w − bw

b ∀w ∈ A.

(5.42)

b It can be verified that (see Exercise 5.12) for all n ≥ 1 and w ∈ A







1 n d (w) = bn w − b(bn−1 w). (5.43) n! Let z be the (possibly infinite) Q-linear span of zk for k ≥ 1. For all b (see Exercise 5.13) such that z ∈ z, we define the derivation δz on A d(a) = ab, d(b) = b2 , and

δz (a) = 0

and

δz (b) = (a + b)z,

(5.44)

b1 → A b 1 by and a map χz : A

  χz (w) = (1 − z) (1 − z)−1 ∗ w

b 1. ∀w ∈ A

(5.45)

Lemma 5.5.3. (i) The vector space z becomes a commutative and associative algebra with respect to the multiplication ◦ defined by z ◦ z 0 = z ∗ z 0 − zz 0 − z 0 z,

∀z, z 0 ∈ z.

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(ii) The Q-linear map

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γ : XQ[[X]] −→ z,

γ(X k ) = zk

∀k ≥ 1,

(5.46)

is an algebra isomorphism for ◦. Moreover, for every f ∈ XQ[[X]] we have f (a) δγ(f ) (a) = 0, δγ(f ) (b) = (a+b) b. (5.47) a Proof. (i) By the definition of ∗ we see that zk ◦ zl = zk ∗ zl − zk zl − zl zk = zk+l . Thus (i) follows immediately. (ii) The first sentence is straightforward (see Exercise 5.14). To prove Eq. (5.47) we only need to assume f (X) = X k for some k ≥ 1. Then γ(X k ) = zk and by the definition in Eq. (5.44) δzk (a) = 0,

δzk (b) = (a + b)zk = (a + b)ak−1 b = (a + b)

f (a) b. a

This completes the proof of the lemma. Proposition 5.5.4. Let z ∈ z. Then we have b 1 ) and (i) χz ∈ Aut(A χz (w) = exp(δt )(w), t = log◦ (1 + z) = −

∞ X

(−1)n

n=1

z ◦n ∈ z. n

(5.48)

b with χz (a) = a and χz (a+b) = (a+b)(1− (ii) χz can be extended to Aut(A) −1 b 0. z) . Further, χz induces an automorphism of A Proof. (i) For any t ∈ z, we have (see Exercise 5.10) exp∗ (t) = (2 − exp◦ (t))−1 .

(5.49) −1

Taking t = log◦ (1 + z) we thus get exp∗ (t) = (1 − z) b 1 we have (see Exercise 5.11) w∈A

. Also, for t ∈ z and

exp(δt )(w) = (exp∗ (t))−1 (exp∗ (t) ∗ w) = (1 − z)((1 − z)−1 ∗ w) = χz (w). (5.50) (ii) By the definition δt (a) = 0 and δt (b) = (a + b)t. So

δt (a + b)w = (a + b) tw + δt (w) .

By induction it is straightforward to show that δtn (b) = (a + b)t∗n for all n ∈ N. Hence by Eq. (5.48) χz (a) = exp(δt )(a) = a and χz (a + b) = exp(δt )(a + b) = (a + b) exp∗ (t) = (a + b)(1 − z)−1 by Eq. (5.49). To show

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b 0 , it suffices to show that χz (w) ∈ A b 0 for χz induces an automorphism of A 0 b since clearly exp(−δt ) is the inverse. Now suppose w = aub. all w ∈ A P∞ Then χz (aub) = aχz (u)χz (b). But χz (b) = b + n=1 (a + b)z n so that b 0. each word ends with b. Hence χz (w) ∈ A This completes the proof of the proposition. For n ≥ 1 let ∂n be the derivation with ∂n (a) = a(a + b)n−1 b,

and ∂n (b) = −a(a + b)n−1 b.

(5.51)

Theorem 5.5.5. (Drivation Relations). For all n ∈ N and words w ∈ A0 , ζ(∂n (w)) = 0. Proof. By Thm. 3.3.21 for all positive integer n one has ζ
(bn ) = ρ ◦ ζ∗ (bn ).

Multiplying ζ(w) on both sides and using the fact that both ζ
and ζ∗ are homomorphisms extending ζ, we get ζ
(bn


w) = ρ ◦ ζ (b ∗

n

∗ w) = ζ
(bn ∗ w),

where the second equality follows from Thm. 3.3.21 again. Recall that we have the isomorphism Eq. (3.14): A1
∼ = A0
[T ]. Let’s denote by reg
the composition of the regularization map A1 ∼ = A0
[T ] → A0 followed by the specialization T = 0. We see that reg
(bn


w) = n!1 reg
(b

w) = n!1 reg
(b) n

n

w = 0.

Hence ζ ◦ reg
(bn ∗ w) = 0 for all positive integer n. So ζ

∞ X

n=1

 reg
(bn ∗ w)un = 0,

(5.52)

b by where u is a formal variable. Define ∆u ∈ Aut(A) ∆u = exp(−du) ◦ χbu .

(5.53)

Now multiplying Eq. (5.43) by un and summing over n yields (after replacing w by v there)   b exp(du)(v) = (1 − bu) (1 − bu)−1 v , ∀v ∈ A. (5.54)




Substituting ∆u (w) for v we see that     (1 − bu) (1 − bu)−1 ∆u (w) = χbu (w) = (1 − bu) (1 − bu)−1 ∗ w .




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Dividing by 1 − bu and then applying reg
we get

∞
X reg
∆u (w) = reg
(bn ∗ w)un .

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n=0

b 0 and Here we use the fact that reg
(1 − bu)−1 )) = 1. Since w ∈ A 0 b both exp(du) and χbu induce automorphisms on A (see Excercise
5.12 and b 0 . Hence reg ∆u (w) = ∆u (w) Prop. 5.5.4(ii)) we see that ∆u (w) ∈ A
b 0 . The theorem now and Eq. (5.52) gives ζ(∆u (w)) = ζ(w) for all w ∈ A follows from the following lemma. b Lemma 5.5.6. As an automorphism of A ∆u = exp

∞ X

(−1)

n ∂n

n=1

n

! n

u

.

(5.55)

Proof. We only need to check their images on the generators a and b are the same. First, by Prop. 5.5.4(ii) ∆u (a) = exp(−du) ◦ χbu (a) = exp(−du)(a) n n X dn (a) n X = (−1)n u = (−1)n abn un = a(1 + bu)−1 . (5.56) n! n=0 n=0 Now by Eq. (5.43) we have exp(du)(a + b) = (a + b)(1 − bu)−1 (see Exercise 5.13) so that ∆u (a + b) = exp(−du) ◦ χbu (a + b)


= exp(−du) (a + b)(1 − bu)−1 = a + b.

(5.57)

Comparing to Prop. 5.5.4(ii) we see that ∆u = εχbu ε

(5.58)

b such that ε(a) = a+b, ε(b) = −b. Therefore, where ε is the involution on A Prop. 5.5.4(i) yields
∆u = ε exp δlog◦ (1+bu) ε. Hence
∆u (a) =ε exp δlog◦ (1+bu) (a + b)
=ε exp δγ(log(1+uX)) (a + b)   log(1 + au) =ε exp (a + b) b a

(by Lemma 5.5.3(ii)) (by Eq. (5.47))

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Multiple Zeta Value Identities

  log(1 + (a + b)u) = exp −a b a+b ! ∞ n−1 X b n n a(a + b) (−1) = exp u n n=1 ! ∞ X ∂ n = exp (−1)n un (a) n n=1

(by Eq. (5.51)),

and
∆u (a + b) = ε exp δlog◦ (1+bu) (a) = ε(a) = a + b = exp

∞ X

(−1)

n=1

n ∂n

n

! n

u

(a + b),

since ∂n (a + b) = 0 for all n ≥ 1. The lemma is now proved. Corollary 5.5.7. For all m ≥ 0 and w = aw0 ∈ A0 , the regularization map satisfies reg
(bm w) =

(−1)m m d (w) = (−1)m a(bm m!


w ).

(5.59)

0

Proof. First note that (1 − bu)−1 = exp
(bu). Taking v = exp(−du)(w) in Eq. (5.54) and multiplying (1 − bu)−1 from the left, we have (1 − bu)−1 w = (1 − bu)−1


exp(−du)(w).

Applying reg
, we get       reg
(1 − bu)−1 w = reg
(1 − bu)−1 reg
exp(−du)(w)   = reg
exp(−du)(w) . By comparing the coefficients of um on both sides, we see easily that the first equality in Eq. (5.59) holds. Now observing that 1/m!dm (a) = abm and 1/m!dm (b) = bm+1 (see Exercise 5.12), we get exp(du)(a) = a(1 − bu)−1 ,

exp(du)(b) = b(1 − bu)−1 .

(5.60)

Taking v = aw0 in Eq. (5.54) and using the first identity of Eq. (5.60) (with u → −u), we get   exp(−du)(w) = exp(−du)(x) exp(−du)(w0 ) = a (1 + bu)−1 w0 .




The second equality in Eq. (5.59) follows by a comparison of the coefficients of um on both sides. We now have completed the proof of the corollary.

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Set Dn (a) = 0 and Dn (b) = an b for all n ≥ 1. For each n ≥ 1, define b → A b as the homogeneous components of the linear maps σn and σn : A degree n of the homomorphisms:     ∞ ∞ X X X Dn X Dn = = σn and σ = τ στ = exp  σ = exp  σn , n n n=1 n=1 n≥1

n≥1

respectively. Lemma 5.5.8. Set ∂ =

P∞

∂n n=1 n .

Then

exp(∂) = σσ −1 .

(5.61)

Proof. Let’s check the two automorphisms’ images on the generators a and a + b are the same. First observe that exp(∂) = ∆−1 . Thus by Eqs. (5.56) and (5.57) exp(∂)(a) = ∆−1 (a) = a(1 − b)−1

and

exp(∂)(a + b) = a + b. (5.62)

Now we set

D=

X Dn . n

n≥1

Then D(a) =

P

n≥1 Dn (a)/n = 0 and X Dn (b) X an b = = (− log(1 − a))b. D(b) = n n n≥1

n≥1

Hence Dn (a) = 0 and Dn (b) = (− log(1 − a))n b and therefore X Dn (a) σ(a) = exp(D)(a) = = a, n!

(5.63)

n≥0

so σ −1 (a) = a. Further, σ(b) = exp(D)(b) =

X Dn (a) n!

n≥0

X (− log(1 − a))n = b = (1 − a)−1 b = exp(∂)(a). n!

(5.64)

n≥0

Similarly σ −1 (b) = (1 − a)b. Thus and

σσ −1 (a) = σ(a) = τ (σ(b)) = a(1 − b)−1

σσ −1 (b) = σ((1 − a)b) = τ σ(a(1 − b))
= τ a(1 − (1 − a)−1 b) = (1 − a(1 − b)−1 )b.

Adding the above two equations, we get σσ −1 (a+b) = a+b. By comparing with Eq. (5.62) this proves the lemma.

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We now prove the next theorem using derivations. Theorem 5.5.9. (Ohno’s relations). Suppose s = (s1 , . . . , sd ) with s1 > 1 and its dual s∗ has depth d0 . Then for any integer n ≥ 0 X X ζ(s + e) = ζ(s∗ + e0 ). (5.65) e0 ∈(N0 )d0 ,|e0 |=n

e∈(N0 )d ,|e|=n

Proof. By Eqs. (5.63) and (5.64) we see that σ(zs1 · · · zsd ) = as1 −1 (1 − a)−1 bas2 −1 (1 − a)−1 b · · · asd −1 (1 − a)−1 b ∞ X X = zs1 +e1 · · · zsd +ed . n=0 e1 +···+ed =n e1 ,...,ed ∈N0

Thus σn (zs1 · · · zsd ) =

X e1 +···+ed =n e1 ,...,ed ∈N0

zs1 +e1 · · · zsd +ed .

Hence, in terms of derivation, Ohno’s relation says that
ζ (σn − σn τ w) = 0 ∀w ∈ A0 , n ≥ 1. By the duality of Eq. (5.4) we only need to show that
ζ (σn − σn )w = 0 ∀w ∈ A0 , n ≥ 1.

(5.66)

But by Lemma 5.5.8 we get σ −σ = (1−exp(∂))σ. Hence Eq. (5.66) follows from Thm. 5.5.5 componentwise. The theorem is now proved. 5.6 Fixed Weight, Height and Depth Relations Another important technique in the study of the MZVs is the use of generating functions, which frequently leads to the generalized hypergeometric functions (see Appendix C for a brief overview of these functions and [517] for a comprehensive treatment). For example, we will use Lemma C.1 to prove the main theorem of this section. First we need to introduce some new notation. Definition 5.6.1. Let s = (s1 , . . . , sd ) ∈ Nd . We define the height of s to be the number of components of s which are greater than 1, denoted by ht(s). For nonnegative integers w, d and h, set  I(w, d, h) = s = (s1 , . . . , sd ) ∈ Nd : |s| = w, ht(s) = h ,  I0 (w, d, h) = s = (s1 , . . . , sd ) ∈ Nd : |s| = w, ht(s) = h, s1 ≥ 2 .

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Theorem 5.6.2. For any nonnegative integers w, d and h, let X G0 (w, d, h) = ζ(s). Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 06/12/16. For personal use only.

s∈I0 (w,d,h)

Then X

G0 (w, d, h)xw−d−h y d−h z h−1 =

1 − exp

w,d,h≥1



P

∞ ζ(n) n=2 n Sn (x, y, z)

xy − z

,

where n

n

n

n

Sn (x, y, z) = x + y − α − β ,

α, β =

x+y±

p

(x + y)2 − 4z . 2

Proof. The basic idea is to treat G0 (w, d, h) as a special point in a one parameter family G0 (w, d, h; t) and then deal with the generating function of the whole family. The advantage is that we can now take derivatives and solve the resulting differential equations. Recall that we have defined the one variable multiple polylog Ls (t) by Eq. (3.20). Further, we have set Ls (t) = 0 if s is empty. Now we define X X G(w, d, h; t) = Ls (t), G0 (w, d, h; t) = Ls (t). s∈I(w,d,h)

s∈I0 (w,d,h)

By Prop. 4.2.1 we get t

d G0 (w, d, h; t) = G(w − 1, d, h − 1; t) − G0 (w − 1, d, h − 1; t) dt + G0 (w − 1, d, h; t), (5.67)

 1 d G(w, d, h; t) − G0 (w, d, h; t) = G0 (w − 1, d − 1, h; t). dt 1−t Consider the generating functions X Ψ (x, y, z; t) =1 + G(w, d, h; t)xw−d−h y d−h z h , w,d,h≥0

Ψ0 (x, y, z; t) =

X

G0 (w, d, h; t)xw−d−h y d−h z h−1 .

w,d,h≥0

Then Eqs. (5.67) and (5.68) yield t

dΨ Ψ − 1 − zΨ0 = + xΨ0 , dt y

(1 − t)

d (Ψ − zΨ0 ) = yΨ. dt

Thus by elimination of Ψ we get  dΨ d2 Ψ0  0 t(1 − t) 2 + 1 − x − (1 − x + y)t + (xy − z)Ψ0 = 1. dt dt

(5.68)

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Clearly Ψ0 (0) = 0. To compute Ψ00 (0), first note that limt→0 Lk (t)/t = 0 unless dp(k) = ht(k) = 1 in which case limt→0 Lk (t)/t = 1. Thus Ψ00 (0) = P∞ w−2 = 1/(1 − x). Setting α + β = x + y and αβ = z, we see that w=2 x  1  1 − 2 F1 (α − x, β − x; 1 − x; t) , Ψ0 (x, y, z; t) = xy − z

since they are both solutions of Eq. (C.3) satisfying the same initial conditions y(0) = 0, y 0 (0) = 1/(1 − x). Thus by Gauss summation theorem Eq. (C.2)   1 Γ(1 − x)Γ(1 − y) Ψ0 (x, y, z; 1) = 1− . xy − z Γ(1 − α)Γ(1 − β) Finally, the theorem follows from the expansion (see Eq. (3.17)) ! ∞ X ζ(n) n Γ(1 − x) = exp γx + x . n n=2

This concludes the proof of the theorem. Corollary 5.6.3. For any nonnegative integers w, d and h, the quantity X G0 (w, d, h) = ζ(s) s∈I0 (w,d,h)

can be expressed as a polynomial of ζ(2), ζ(3), . . . with rational coefficients. Proof. This follows from the theorem immediately. Corollary 5.6.4. For any positive integer m,
ζ {2}m =

π 2m . (2m + 1)!

Proof. In the generating function Ψ0 (x, y, z; 1) if we set x = y = 0 then the only nonzero terms are given by d = h and w = 2d meaning that Ψ0 (x, y, z; 1) =

∞ X s=1


ζ {2}s z s−1 .

By the theorem we get ∞ X m=1

ζ {2}

m




z

m−1

1 = z

∞ X ζ(2n) exp − (−z)n n n=1

1 = z

∞ Y k=1

!

∞ X (−z/k 2 )n exp − n n=1

! −1

!

! −1

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Multiple Zeta Functions, Multiple Polylogarithms, and Their Special Values ∞  Y

! z  1+ 2 −1 k k=1   X √ ∞ 1 sinh π z π 2m √ = −1 = z m−1 . z (2m + 1)! π z m=1 1 = z

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The corollary follows from this immediately. 5.7 Zagier’s 2-3-2 Formula of MZVs In this section we will present a 2-3-2 formula of the MZVs, i.e., a formula which involves the MZVs of the form L(a, b) := ζ({2}a , 3, {2}b ),

a, b ∈ N0 .

(5.69)

Zagier first proved this formula which has an important application in Brown’s proof of the Deligne-Ihara Conjecture and part of Hoffman’s Basis Conjecture (see Conjection 3.6.1) in [102]. The key step is to express the quantities in Thm. 5.7.4 using the hypergeometric functions. The formula in the theorem then follows from some transformation relations of the hypergeometric functions. Lemma 5.7.1. We have the generating function FL (x, y) :=

∞ X

(−1)a+b+1 L(a, b)x2a+1 y 2b+2

a,b=0

  sin πx d y, −y, z F , 3 2 π dz z=0 1 + x, 1 − x     α1 ,α2 ,α3 2 ,α3 where the hypergeometric functions 3 F2 α1β,α = F ; 1 is 3 2 β1 ,β2 1 ,β2 defined in Appendix C. =

Proof. Using the definition of the MZVs and the Pochhammer symbol Eq. (1.13), we have    ∞ Y X x2 1 Y y2 FL (x, y) = − xy 2 1− 2 1 − l n3 k2 n=1 l>n

n>k>0

∞ sin πx X 1 (−y)n (y)n = π n=1 n (1 − x)n (1 + x)n

which can be easily transformed into the formula in the lemma.

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163

For any nonnegative integers a and b, we define      a+b+1 X 2n 2n 1 n R(a, b) := 2 (−1) − 1− n 2a + 1 4 2b + 2 n=1
a+b+1−n × ζ(2n + 1)ζ {2} . Lemma 5.7.2. The generating function ∞ X FR (x, y) := (−1)a+b+1 R(a, b)x2a+1 y 2b+2 a,b=0

satisfies the relation π · FR (x, y) 1 X = ψ(1 + x) + ψ(1 − x) − ψ(1 + κx + λy) sin πx 2 κ,λ=±1    κx + λy  sin πy X κλ ψ 1 + − ψ(1 + κx + λy) . (5.70) − 2 sin πx 2 κ,λ=±1

Proof. By the definition and Cor. 5.6.4, we have  n−1 ∞ X  2n  X (−1)k π 2k x2k+2n−2b−1 y 2b+2 FR (x, y) = 2 ζ(2n + 1) 2b + 2 (2k + 1)! k,n=1

b=0

  n−1  1 X 2n 2k+2n−2a 2a+1 − 1− n y x 4 a=0 2a + 1 

=

∞   sin πx X ζ(2n + 1) (x + y)2n + (x − y)2n − 2x2n π n=1

−

∞   sin πy X  1 1 − n ζ(2n + 1) (x + y)2n − (x − y)2n π n=1 4

 sin πy   sin πx  A(x + y) + A(x − y) − 2A(x) − B(x + y) − B(x − y) , π π where, for all |z| < 1 ∞ ∞  X X 1 z A(z) = ζ(2n+1)z 2n , B(z) = 1− n ζ(2n+1)z 2n = A(z)−A( ). 4 2 n=1 n=1 =

Moreover, using the expansion Eq. (1.5) of the digamma function ψ(x), we can see easily that 1 1 A(z) = ψ(1) − ψ(1 + z) − ψ(1 − z). 2 2 This finishes the proof of the lemma.

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We also need a result concerning the digamma function ψ(x) (see p. 2). Lemma 5.7.3. When z → 0 we have Γ(x) = 1 + ψ(x)z + o(z). Γ(x − z) Proof. By L’Hˆ opital’s Rule, we have Γ(x) − Γ(x − z) Γ0 (x − z) Γ0 (x) lim = lim 0 = = ψ(x). z→0 z→0 Γ (x − z)z + Γ(x − z) Γ(x − z)z Γ(x) The lemma follows immediately. We are now ready to prove Zagier’s 2-3-2 formula of the MZVs. Theorem 5.7.4. For any a, b ∈ N0 , we have L(a, b) = R(a, b). Namely,      a+b+1 X 2n 1 2n − 1− n ζ({2}a , 3, {2}b ) = 2 (−1)n 2b + 2 4 2a + 1 n=1
a+b+1−n × ζ(2n + 1)ζ {2} . (5.71) Proof. First we observe that 1 1 (y)n (−y)n = (y)n (1 − y)n + (1 + y)n (−y)n . 2 2 Thus       y, 1 − y, z 1 1 + y, −y, z y, −y, z 1 + 3 F2 . = 3 F2 3 F2 2 1 + x, 1 − x 2 1 + x, 1 − x 1 + x, 1 − x (5.72) Because of the symmetry y ↔ −y on the right-hand side of Eq. (5.72), we only need to consider the first 3 F2 -term. Applying the transformation formula Eq. (C.6) with α1 = y, α2 = z, α3 = 1−y, β1 = 1+x and β2 = 1−x, we get     y, 1 − y, z Γ(1 + x)Γ(1 + x − y − z) y, y − x, z = 3 F2 3 F2 Γ(1 + x − y)Γ(1 + x − z) y − x + z, 1 − x 1 + x, 1 − x Γ(1 + x)Γ(1 − x)Γ(y − x + z − 1)Γ(1 − z) + Γ(y)Γ(z)Γ(y − x)Γ(2 − y − z)   1 + x − y, 1 + x − z, 1 − z × 3 F2 . (5.73) 2 + x − y − z, 2 − y − z Applying the transformation Eq. (C.7) to the first 3 F2 -term on the righthand side of Eq. (5.73) with α1 = y, α2 = y − x, α3 = z, β1 = y − x + z and β2 = 1 − x, we get   y, 1 − y, z Γ(1 + x) Γ(1 − x) Γ(1 − x − y) F = 3 2 1 + x, 1 − x Γ(1 + x − z) Γ(1 − x − z) Γ(1 − x − y + z)

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  −x + z, z, z Γ(1 + x − y − z) F 3 2 y − x + z, 1 − x − y + z Γ(1 + x − y) xΓ(x)Γ(1 − x) Γ(y − x − 1 + z) + Γ(y)Γ(2 − y − z) Γ(y − x)   1 + x − y, 1 + x − z, 1 − z Γ(1 − z) F × 3 2 2 + x − y − z, 2 − y − z Γ(z)

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×

= [1 + ψ(1 + x)z][1 + ψ(1 − x)z][1 − ψ(1 − x − y)z][1 − ψ(1 + x − y)z]   x sin πy 1 + x − y, 1 + x, 1 z + F + o(z) 3 2 2 + x − y, 2 − y 1 − y sin πx y − x − 1

as z → 0 by Lemma 5.7.3. Here, we have also used the reflection formula π Γ(t)Γ(1 − t) = sin πt for the Γ-product coefficient of the second 3 F2 -term. Now taking α = 1 + x − y and β = 1 + x in the summation formula of Prop. C.3, we get   d y, 1 − y, z = ψ(1+x)+ψ(1−x)−ψ(1+x−y)−ψ(1−x−y) 3 F2 dz z=0 1 + x, 1 − x   x − y x + y i sin πy h · ψ(1 + x − y) − ψ(1 − x − y) − ψ 1 + +ψ 1− . − sin πx 2 2 (5.74) Combining Eqs. (5.72) and (5.74), we see that FL (x, y) = FR (x, y) which yields the theorem immediately. If k, n1 , . . . , nd ∈ N, we have defined the following shuffle-regularized MZVs (see Prop. 3.2.4): ζk (n1 , . . . , nd ) := (−1)d ζ
(an1 −1 b · · · and −1 bak ). Corollary 5.7.5. For any n ∈ N, we have

n X

ζ1 {2}n = 2 (−1)i ζ(2i + 1)ζ {2}n−i . i=1

Proof. By Thm. 4.1.14 we have d−1 X

ζ1 {2}d = 2 ζ {2}a , 3, {2}d−a−1 a=0

=4

d X i=1

i

d−i

(−1) ζ(2i + 1)ζ {2}

d−1
X a=0



    2i 1 2i − 1− i 2d − 2a 4 2a + 1

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by Thm. 5.7.4. So the corollary follows from the well-known combinatorial identities:   i−1  i−1  X X 2i 2i 2i−1 =2 − 1 and = 22i−1 , 2b + 2 2a + 1 a=0 b=0

which can be proved, for e.g., by using generating functions or the WZmethod [472]. 5.8 An Exotic Shuffle Relation Using the shuffle relations of the MZVs we can prove the following identity. Usually, the shuffle product is defined on the algebra of words. But in the statement of the theorem (but not in the proof) we treat the numbers 1,2 and 3 as letters and then extend the zeta function ζ linearly over Z. Theorem 5.8.1. For any nonnegative integers m and n,  
2n + m π 4n+2m m n . ζ {2} {3, 1} = (2n + 1) · (4n + 2m + 1)! m




(5.75)

Proof. The case n = 0 is just Cor. 5.6.4: for every positive integer r ζ({2}r ) =

π 2r . (2r + 1)!

(5.76)

Applying the evaluation map ζ defined in Prop. 3.2.4 to the two equations in Prop. 3.2.6 and using Eq. (5.76) we see that n X k=1−n n X k=−n

m X (−1)k k 2m−1 π 4n−2 = 4n x2n−1,n−j · a2m−1,j , (2n + 2k − 1)!(2n − 2k + 1)! j=1

(5.77)

m X (−1)k k 2m π 4n n =4 x2n,n−j · a2m,j , (2n + 2k + 1)!(2n − 2k + 1)! j=1

(5.78)

R 1 n−j R 1 n−j where x2n−1,n−j = − 0 T2n−1 and x2n,n−j = 0 T2n . Observe that ζ({2}r ) = (−1)r ζ((ab)r ) and ζ({3, 1}r ) = ζ((a2 b2 )r ), so it is not too hard to see that x2n−1,n−j = ζ({2}2j−1


{3, 1}

n−j

),

x2n,n−j = ζ({2}2j


{3, 1}

n−j

).

Note that for arbitrary fixed n, x2n−1,n−j (resp. x2n,n−j ) are recursively defined by Eq. (5.77) (resp. Eq. (5.78)) when we take m = 1, . . . , n − 1.

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Thus Thm. 5.8.1 is quickly reduced to the following combinatorial identities: for all m, n ≥ 1, we have   n X 4n − 1 (−1)k k 2m−1 2n − 2k + 1 2n + 2k − 1 k=1−n

   j m X X 4n−j (−1)k k 2m−1 2n − 1 2j − 1 , 2n − 2j + 1 2j − 1 j−k j=1 k=1−j   n X 4n + 1 (−1)k k 2m 2n − 2k + 1 2n + 2k + 1 =

(5.79)

k=−n

=

   j m X X 4n−j (−1)k k 2m 2n 2j . 2j j−k 2n − 2j + 1 j=1

(5.80)

k=−j

Now observe that we may replace the outer sum of the right-hand side in P∞ both Eq. (5.79) and Eq. (5.80) by j=1 by the following computation: if j > m then    2m−1 X   j j X d k 2m−1 2j − 1 k k 2j − 1 (−1) k = x (−1) x j−k dx j − k x=1 k=1−j k=1−j  2m−1 d 1−j = (−1) x x1−j (1 − x)2j−1 = 0. dx x=1 Similarly, if j > m then j X k=−j

k 2m

(−1) k



2j j−k

 = 0.

We therefore only need to prove that for all m, n ≥ 1   n X 4n − 1 (−1)k k m 2n − 2k + 1 2n + 2k − 1 k=1−n

   j ∞ X X 4n−j (−1)k k m 2n − 1 2j − 1 , 2n − 2j + 1 2j − 1 j−k j=1 k=1−j   n X 4n + 1 (−1)k k m 2n − 2k + 1 2n + 2k + 1 =

(5.81)

k=−n

=

   j ∞ X X 4n−j (−1)k k m 2n 2j . 2n − 2j + 1 2j j−k j=1 k=−j

(5.82)

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In fact, we only need to consider the case m is odd in Eq. (5.81) and m is even in Eq. (5.82). It is a
well-known fact that xQ[x] has two bases over Q: {xm : m ≥ 1} x and { m : m ≥ 1}. So Thm. 5.8.1 follows immediately from the following Prop. 5.8.2. Note that we will exchange the indices j and k on the righthand side. Proposition 5.8.2. For all m, n ≥ 1, we have    n X (−1)k k 4n − 1 2n − 2k + 1 m 2n + 2k − 1 k=1−n

∞ X k X

    2n − 1 2k − 1 4n−k (−1)j j , 2n − 2k + 1 m 2k − 1 k−j k=1 j=1−k    n X k 4n + 1 (−1)k 2n − 2k + 1 m 2n + 2k + 1 =

(5.83)

k=−n

    ∞ X k X 4n−k (−1)j j 2n 2k = . 2n − 2k + 1 m 2k k−j

(5.84)

k=1 j=−k

Proof. We first break the inner sum on the right-hand side of Eq. (5.83) as follows:    k X j 2k − 1 (−1)j m k−j j=1−k

   k−1    k X X 2k − 1 2k − 1 j j m−j m + j − 1 = (−1) + (−1) . m k−j m j+k j=1 j=1

Re-indexing and using the Chu–Vandermonde identity (see [472, p. 182]) we change the above to    k−2    k−m X X −m − 1 2k − 1 −m − 1 2k − 1 (−1)m + (−1)m−1 j k−2−j j k−m−j j=0 j=0     2k − m − 2 2k − m − 2 . = (−1)m + (−1)m−1 k−m k−2 Note that the sum is 0 when k ≥ m, however, when k < m only the first term in the sum is always 0. By denoting the left-hand (resp. right-hand) side of Eq. (5.83) by L(m, n) (resp. R(m, n)) we get       n X (−1)m 4n−k 2n − 1 2k − m − 2 2k − m − 2 R(m, n) = − . (2n − 2k + 1) 2k − 1 k−m k−2 k=1 (5.85)

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Set    k 4n − 1 (−1)k , 2n − 2k + 1 m 2n + 2k − 1    (−1)m+1 4n−k 2n − 1 2k − m − 2 F1 (n, k) := , 2n − 2k + 1 2k − 1 k−2    (−1)m 4n−k 2n − 1 2k − m − 2 . F2 (n, k) := 2n − 2k + 1 2k − 1 k−m

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F0 (n, k) :=

Then clearly limk→±∞ Fi (n, k) = 0 for each n and  X X F1 (n, k) + F2 (n, k) . L(m, n) = F0 (n, k), R(m, n) = k∈Z

k∈Z

Now by the WZ-method (see [472]) we can find functions Gi (n, k) such that all of the three functions Fi above satisfy the following: c3 Fi (n + 3, k) − c2 Fi (n + 2, k) + c1 Fi (n + 1, k) − c0 Fi (n, k) = Gi (n, k + 1) − Gi (n, k),

∀k ∈ Z,

(5.86)

where limk→±∞ Gi (n, k) = 0 for each fixed n and

c0 =32n(4n + 1)(2n + 1)(4n + 3)(n + 1)[144n4 − 168n3 m

+ 40n2 m2 + 6nm3 + 1128n3 − 876n2 m + 97nm2 + 10m3 + 3188n2 − 1457nm + 51m2 + 3868n − 775m + 1704],

c3 =(2n + 3)(n + 3)(n + 1)(2n − 2m + 7)(n − m + 3)[144n4

− 168n3 m + 40n2 m2 + 6nm3 + 552n3 − 372n2 m + 17nm2 + 4m3 + 668n2 − 209nm − 6m2 + 300n − 26m + 40].

Similarly we can find explicit expression of c1 and c2 (see Exercise 5.15). Now summing up Eq. (5.86) for −∞ < k < ∞ we find that both L(m, n) and R(m, n) satisfy the same recurrence relation c3 A(n + 3) = c2 A(n + 2) − c1 A(n + 1) + c0 A(n).

Furthermore it is easy to check by using Eq. (5.83) that for all m ≥ 1 L(m, 1) = R(m, 1) = −δm,1 ,

L(m, 2) = R(m, 2) = −5δm,1 − (−1)m ,

where δm,1 is the Kronecker symbol. When n = 3 direct computation using Eq. (5.85) shows that   if m = 1, −16   0 if m = 2, L(m, 3) = R(m, 3) =  40/3 if m = 3,     m (−1) (m − 52/3) if m ≥ 4.

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This implies that Eq. (5.83) is true. The proof of Eq. (5.84) can be carried out by exactly the same method as above so we leave the details to the interested reader. This completes the proof of the proposition. 5.9 Period Polynomial Relations of Double Zeta Values In Remark 3.4.7 concerning Conjecture 3.4.6 about the dimensions of MZV space we mentioned that MZVs are expected to be tied up with the modular forms. In this section we will explore this important relation in some details. We will always assume k to be an even positive integer throughout this section, except when we point out otherwise. 5.9.1

Double Zeta Space

Recall that the double zeta values satisfy the following stuffle and shuffle relations (see Eq. (5.16)): ζ(j)ζ(w − j) = ζ(j, w − j) + ζ(w − j, j) + ζ(w), w−1 X r − 1  r − 1  + ζ(r, w − r). ζ(j)ζ(w − j) = j−1 w−j−1 r=2

(5.87) (5.88)

for 2 ≤ j ≤ w/2. Equating the right-hand side of these two relations we get w−1 X r − 1  r − 1  + ζ(r, w − r). ζ(j, w − j) + ζ(w − j, j) + ζ(w) = j−1 w−j−1 r=2 (5.89) We define the formal double zeta space Dw to be the Q-vector space generated by the formal symbols Zr,s , Pr,s and Zw subject to the above two sets of relations, with ζ(r, s), ζ(r)ζ(s) and ζ(w) replaced by Zr,s , Pr,s and Zw , respectively, by formally allowing r and s to be 1. Clearly we can also write Dw =

QhZr,w−r , Zw : 1 ≤ r < wi , Qhrelations (5.89)i

with w generators and bw/2c relations, which shows that dim Dw ≥

w . 2

We will show in fact the equality holds in Thm. 5.9.11.

(5.90)

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Theorem 5.9.1. Let w be a positive even integer. Then X X 3 1 Zr,w−r = Zw . Zr,w−r = Zw , 4 4 1 0. Define X (α)k1 kd ! 1 Z(s; α) = . k1 ! (α)kd +1 (k1 + α)s1 · · · (kd + α)sd k1 >···>kd >0

where (α)k is the Pochhammer symbol defined by Eq. (1.13). Then for any integer k ≥ 0 admissible s = (s1 , . . . , sd ) ∈ Nd , show that X X Z(s01 + ε01 , . . . , s0` + ε0` ; α). Z(s1 + ε1 , . . . , sd + εd ; α) = ε1 +···+εd =k ε1 ,...,εd ∈N0

ε01 +···+ε0` =k ε01 ,...,ε0d ∈N0

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5.7. One may use the following idea to give an alternative elementary proof of Thm. 5.4.1. First, rewrite the double sum as

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w−1 X

∞ X

k=2 m,n=1

2k . + m)k

nw−k (n

Then break the sum into two cases m = n and m 6= n. Show that the first case yields (w − 2)ζ(w) and the second yields 3ζ(w) by summing over k first and then using telescoping series. 5.8. Let w ≥ 2 be a positive integer. Using the generating function of the double zeta values to show the following restricted sum formulas:   X X X X   ζ(k, l) = 1 − − ζ(k, l) if w ≡ 0 (3), 3 k odd k≡3 (6) k≡4 (6) k≡5 (6)   X X X X   ζ(k, l) = 1 + − ζ(k, l) if w ≡ 1 (3), 3 k≡3 (6)

k≡4 (6)

k even

k≡5 (6)

X 1 X 1 ζ(w) − ζ(k, l) = ζ(k, l) 6 3

if w ≡ 2 (3),

k odd

k≡4 (6)

where in the sums we always assume k ≥ 2, l ≥ 1 and k + l = w. 5.9. Show that the commutator of two derivations is a derivation. 5.10. For any t ∈ z define a power series f (u) ∈ z[[u]] by X t◦n un . f (u) = exp◦ (tu) − 1 = n! n≥1

(i) Show that d du



1 1 − f (u)



 =t∗

1 1 − f (u)



by using the fact that f 0 (u) = t ◦ (1 + f (u)). (ii) Show that F (u) = (1−f (u))−1 satisfies F 0 (u) = z ∗F (u) and F (0) = 1. Consequently, one has exp∗ (tu) = (2 − exp◦ (tu))−1 . b 1. 5.11. Let t ∈ z and w ∈ A (i) Show that for any z ∈ z δt (zw) = (t ◦ z)w + z(t ∗ w).

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(ii) Show that for any positive integer n   X n n δt (zw) = (t◦α ◦ z)(t∗β ∗ w). α Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 06/12/16. For personal use only.

α+β=n,α,β≥0

(iii) Show that exp(δt )(w) = (exp∗ (t))−1 (exp∗ (t) ∗ w).




5.12. Show that d : A → A defined by d(w) = b w − bw is a derivation on A such that dm (a)/m! = abm and dm (b)/m! = bm+1 for all m ≥ 1. b 0 and then verify Eq. (5.43) Further, show that d induces a derivation on A by induction. In particular, and

exp(du)(a) = a(1 − bu)−1 , exp(du)(b) = b(1 − bu)−1 , exp(du)(a + b) = (a + b)(1 − bu)−1 .

b1 → A b 1 defined by δ(w) = z ∗ w − zw is a derivation 5.13. Show that δz : A 1 b on A for every z ∈ z, and these derivations all commute. Furthermore, δz b with extends to a derivation on all of A, δz (a) = 0

and

δz (b) = (a + b)z.

5.14. Show that γ defined by Eq. (5.46) provides an algebra isomorphism '

γ : XQ[[X]] −→ (z, ◦). 5.15. Use your favorite computer algebra system to find c1 and c2 in the proof of Prop. 5.8.2. One can download the free computer programs for the WZ-method from the internet after searching “WZ-method”. 5.16. Use appropriate specializations of the generating function for double and triple zeta values to show that for all w ≥ 1   X 3a−1 2b − 2a+b−1 − 2a−1 ζ(a, b, w − a − b) a≥2,b≥1,a+b · · · > k1d there is some i (i = 0, . . . , d) 1 such that k11 > · · · k1i > 0 > ki+1 > · · · > k1d . Hence X 1 (−1)|s| s1 · · · kdsd k 1 1 1 k1 >···> kd 06=|kj |∈Z≤M ∀j=1,...,d

=

d X i=0

X 1 k1

>···> k1 >0> k i

1 i+1

>···> k1

d

1 k1s1 · · · kdsd

06=|kj |∈Z≤M ∀j=1,...,d

=

d X (−1)s1 +···+si HM (si , . . . , s1 )HM (si+1 , . . . , sd ) i=0

where HM is defined by (3.18). Clearly HM satisfies the stuffle relation so that d X (−1)s1 +···+si HM (si , . . . , s1 )HM (si+1 , . . . , sd ) i=0

=

d X i=0

=

d X i=0


(−1)s1 +···+si HM (si , . . . , s1 ) ∗ (si+1 , . . . , sd )
(−1)s1 +···+si ζ∗ (si , . . . , s1 ) ∗ (si+1 , . . . , sd ) + O(T −1 logA (T ))

by Lemma 3.3.19. Here T = log M + γ, γ is Euler’s constant, and A is some contant. We also have used the same proof as in Prop. 6.1.3 to reduce the expression in the stuffle by using only admissible compositions for which Lemma 3.3.19 applies. Since ζ∗ satisfies the stuffle relation, we finally obtain the expression d X (−1)s1 +···+si ζ∗ (si , . . . , s1 ; T )ζ∗ (si+1 , . . . , sd ; T ) + O(T −1 logA (T )). i=0

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By Prop. 6.1.3 we see that the sum above is also equal to ζ∗S(s).

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This proposition clearly implies the following corollary. Corollary 6.1.5. The ∗-symmetrized MZVs satisfy the stuffle relations. 6.2 Identities Involving SMZVs In this subsection, we will look at some relations among the SMZVs that are analogs of those for ordinary MZVs. First, we want to present a kind of sum formula, which requires a preliminary combinatorial lemma. For any w, d, i ∈ N with 1 ≤ i ≤ d ≤ w − 1, we set Jw,d,i = {(s1 , . . . , sd ) ∈ Nn | s1 + · · · + sd = w, si ≥ 2}.

Lemma 6.2.1. Let f be a multiple integer variable function satisfying the stuffle relations, namely, f (t1 )f (t2 ) = f (t1 ∗t2 ) for any tj ∈ Ndj (j = 1, 2). Set X Sw,d,i := f (s1 , . . . , sd ). (s1 ,...,sd )∈Jw,d,i

If f (s) = 0 for all s ∈ N then for any w, d, i ∈ N with 2 ≤ i + 1 ≤ d ≤ w − 1, we have (d − i)Sw,d,i + iSw,d,i+1 + (w − d)Sw,d−1,i = 0. Proof. By stuffle relation 0 = f (l)f (s1 , . . . , sd−1 ) =

d X

d−1 X f (s1 , . . . , sj−1 , l, sj , . . . , sd−1 )+ f (s1 , . . . , sj−1 , sj +l, sj+1 , . . . , sd−1 ).

j=1

j=1

Summing this over all (s1 , . . . , sd−1 , l) ∈ Jw,d,i gives the desired recurrence relations. Indeed, the map (s1 , . . . , sd−1 , l) 7→ (s1 , . . . , sj−1 , l, sj , . . . , sd−1 )

defined on Jw,d,i is a bijection to Jw,d,i+1 for 1 ≤ j ≤ i and is surjection onto Jw,d,i for i < j ≤ d. Moreover, under the map Jw,d,i × {1, . . . , d − 1} −→ Jw,d−1,i
(s1 , . . . , sd−1 , l), j 7−→ (s1 , . . . , sj−1 , sj + l, sj+1 , . . . , sd−1 ),

the pre-images of each (s01 , . . . , s0d−1 ) ∈ Jw,d−1,i form a set of cardinality X (s0j − 1) + (s0i − 2) = w − d. 1≤j≤d−1 j6=i

This completes the proof of the lemma.

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The identity in the next theorem is often called a sum formula. In the proof we will use Thm. 6.3.2 from next section, which one can check does not depend on the result of this section. Theorem 6.2.2. For any s = (s1 , . . . , sd ) ∈ Nd and w, i ∈ N with 1 ≤ i ≤ d ≤ w − 1, we have, modulo ζ(2),     X w−1 S i−1 d w−1 ζ∗ (s) ≡ (−1) + (−1) ζ(w). (6.4) i−1 d−i |s|=w, si ≥2

Proof. We prove the theorem by backward induction on the depth d. When d = w − 1 the right-hand side of Eq. (6.4) is 0 when w is even and is       w−1 w−1 w (−1)i−1 + ζ(w) = (−1)i−1 ζ(w) i−1 i i

when w is odd. Now consider the left-hand side of Eq. (6.4). Since ζ∗S satisfies the stuffle relation by Cor. 6.1.5 we may apply Lemma 6.2.1 with f = ζ∗S. By Cor. 5.5.7 we see that   j j m m m j+m reg
(xm x x ) = (−1) x (x x ) = (−1) x0 xm+j . 0 1 1 0 1 1 1 m




Therefore ζ
({1}

i−1

w−i−1

, 2, {1}

 w−1 ζ(2, {1}w−2 ) ) = (−1) i−1   i−1 w − 1 = (−1) ζ(w) i−1 i−1



by the duality relation (5.3). Similarly

ζ
({1}w−i−1 , 2, {1}i−1 ) = (−1)w−i−1



 w−1 ζ(w). i

Thus by the definition and Thm. 6.3.2 S Sw,w−1,i = ζ∗S({1}i−1 , 2, {1}w−i−1 ≡ ζ
({1}i−1 , 2, {1}w−i−1 )

≡ ζ
({1}i−1 , 2, {1}w−i−1 ) + (−1)w ζ
({1}w−i−1 , 2, {1}i−1 )   i−1 w ≡ (−1) ζ(w) (mod ζ(2)). i

So the left-hand side of Eq. (6.4) agrees with the right-hand side. Now we assume that Eq. (6.4) holds for some d ≤ w−1. By Lemma 6.2.1 (d − w)Sw,d−1,i = (d − i)Sw,d,i + iSw,d,i+1     w−1 w−1 = (d − i)(−1)i−1 + (−1)d ζ(w) i−1 d−i

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    w−1 w−1 + i(−1)i + (−1)d ζ(w) i d−i−1      w−1 w−1 = (−1)i−1 (d − i) + (w − d + i)(−1)d ζ(w) i−1 d−i−1      w−1 w−1 + (−1)i (w − i) + i(−1)d ζ(w) i−1 d−i−1     w−1 w−1 = (d − w)(−1)i−1 + (−1)d−1 ζ(w). i−1 d−i−1 Thus Eq. (6.4) holds for d − 1 and the theorem is proved. Recall that the multiple zeta X star values are defined by ? ζ (s) = k1−s1 · · · kd−sd . k1 ≥···≥kd >0

Lemma 6.2.3. For positive integers k and n with k ≥ 2, we have the duality in height 1: ζ
({1}d−1 , k) = (−1)d−1 ζ ? (k, {1}d−1 ). Proof. Set (. . . , i, {1}−1 , j, . . .) = (. . . , i + j − 1, . . .). By Cor. 5.5.7, we have ζ
({1}d−1 , k) = (−1)d+k ζ
(x0 (xd−1 xk−2 x1 )) (word depth=k − 1) 1 0 X e = (−1)d+k (e1 + 1)ζ(x0 x1k−1 · · · x0 xe11 +1 )




e1 +···+ek−1 =d−1 ei ≥0 (i=1,2,...,k−1)

e

X

= (−1)d+k

e1 +···+ek−1 =d−1 ei ≥0 (i=1,2,...,k−1)

X

= (−1)d−1

(e1 + 1)ζ(x0e1 +1 x1 · · · x0k−1 x1 )

(by Eq. (5.3))

(e1 + 1)ζ(2 + e1 , 1 + e2 , . . . , 1 + ek−1 )

e1 +···+ek−1 =d−1 ei ≥0 (i=1,2,...,k−1)

= (−1)d−1

d X

X

ζ(t + 1 + e1 , 1 + e2 , . . . , 1 + ek−1 )

t=1 e1 +···+ek−1 =d−t ei ≥0 (i=1,2,...,k−1)

= (−1)d−1

d X t=1

X

ζ(k + e01 , 1 + e02 , . . . , 1 + e0t ).

e01 +···+e0t =d−t e0j ≥0 (j=1,2,...,t)

by Thm. 5.5.9 of Ohno’s Relations. It is not hard to see that the last sum can be expressed as (see Eq. (1.10) and Exercise 3.17) X (−1)d−1 ζ(k ◦ |1 ◦ ·{z · · ◦ 1} ) = (−1)d−1 ζ ? (k, {1}d−1 ), (d − 1)-times

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where ◦ is either “+” or “,”. The lemma follows immediately.

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The SMZVs have the following height 1 duality relations. Theorem 6.2.4. For any k, n ∈ N, we have

S S ζ
(k, {1}d−1 ) − ζ
(d, {1}k−1 )

is equal to ζ(2) times a polynomial of the Riemann zeta values with rational coefficients. Proof. The claim is trivial if k = 1 or d = 1 so we may assume k, d ≥ 2. By Lemma 6.2.3, we have S S ζ
(k, {1}d−1 ) − ζ
(d, {1}k−1 )

= ζ(k, {1}d−1 )+(−1)k ζ ? (k, {1}d−1 )−ζ(d, {1}k−1 )−(−1)d ζ ? (d, {1}k−1 ).

Taking z = 0 in Thm. 5.6.2, we get X 1− ζ(k + 1, {1}d−1 )xk y d k,d≥1

X  xd + y d − (x + y)d Γ(1 − x)Γ(1 − y) = exp ζ(d) = . d Γ(1 − x − y)

(6.5)

d≥2

Hence by Eq. (1.5) we have  X ζ(k, {1}d−1 ) − ζ(d, {1}k−1 ) xk−1 y d−1 k,d≥2

 =

1 1 − y x

 1−

 Γ(1 − x)Γ(1 − y) +ψ(1 − x) − ψ(1 − y). Γ(1 − x − y)

On the other hand, by Cor. 10.7.6  X k ? d−1 d ? k−1 (−1) ζ (k, {1} ) − (−1) ζ (d, {1} ) xk−1 y d−1 k,d≥2

= −ψ(x) + ψ(y) − π(cot(πx) − cot(πy))

Γ(1 − x)Γ(1 − y) . Γ(1 − x − y)

Combining with the well-known reflection formula (1.6) of the digamma function, we have  X S S ζ
(k, {1}d−1 ) − ζ
(n, {1}k−1 ) xk−1 y d−1 k,d≥2

 =

1 1 − y x

 1−

 Γ(1 − x)Γ(1 − y) +ψ(1 − x) − ψ(1 − y) Γ(1 − x − y)

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Γ(1 − x)Γ(1 − y) − ψ(x) + ψ(y) − π(cot(πx) − cot(πy)) Γ(1 − x − y)   Γ(1 − x)Γ(1 − y) (ψ(1 − x) − ψ(1 + x) − ψ(1 − y) + ψ(1 + y)) = 1− Γ(1 − x − y)   Γ(1 − x)Γ(1 − y) X = −2 1 − ζ(2l)(x2l−1 − y 2l−1 ), Γ(1 − x − y) l≥1

by Eq. (1.5). Since all the coefficients of Γ(1 − x)Γ(1 − y)/Γ(1 − x − y) can be expressed as polynomials of the Riemann zeta values with rational coefficients according to Eq. (6.5), Thm. 6.2.4 follows immediately. 6.3 Drinfeld Associator and SMZVs To investigate other useful properties of the SMZVs it is more enlightening to explore their connections to the Drinfeld associator discussed in Sec. 4.1. Recall that Φ
= π(Φ(a, b)) where Φ(a, b) is the Drinfeld associator and π : R → R/Ra. Here, for simplicity, we rewrite x0 = a and x1 = b. Then it follows from Thm. 4.1.11 that Φ
[w] = (−1)d ζ
(w; 0)

for all w ∈ A1
.

As before we put ζ
(s1 , . . . , sd ) := (−1)d ζ
(as1 −1 b · · · asd −1 b; 0) and ζ∗ (s1 , . . . , sd ) := (−1)d ζ∗ (as1 −1 b · · · asd −1 b; 0) as the constant terms of the regularized MZVs. Theorem 6.3.1. For every word w = as1 −1 b · · · asd −1 b ∈ A1
, the SMZV S ζ
(s1 , . . . , sd ) = (−1)d (Φ−1 bΦ)[bw].

Proof. By the duality relation in Eq. (4.6) for Φ, for all i = 0, . . . , d Φ−1 [bas1 −1 · · · basi −1 ] = Φ[abs1 −1 · · · absi −1 ]

= (−1)s1 +···+si −i ζ
(abs1 −1 · · · absi −1 ; 0) = (−1)s1 +···+si −i ζ
(asi −1 b · · · as1 −1 b; 0)

by the duality relation of MZVs (see Thm. 5.1.2). Therefore (Φ−1 bΦ)[bas1 −1 b · · · asd −1 b] =

d X i=0

Φ−1 [bas1 −1 · · · basi −1 ]Φ[asi+1 −1 b · · · asd −1 b]

d X S = (−1)s1 +···+si +d ζ
(si , . . . , s1 )ζ
(si+1 , . . . , sd ) = (−1)d ζ
(s) i=0

by Prop. 6.1.3, as desired.

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We can also use this Hopf algebra set-up to give another proof of Cor. 6.1.5. For ] = or ∗, we set X Ψ] (a, b) = 1 + ζ] (w; 0)w.




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w∈X∗ b

Define inv(1) = 1 and (6.6) inv(as1 −1 b . . . asr −1 b) = (−1)s1 +···+sr asr −1 b . . . as1 −1 b for all s1 , . . . , sr ∈ N. Since Ψ∗ is group-like for ∆∗ , one can check (see inv inv inv Exercise 6.3) that Ψinv ∗ is also group-like, namely, ∆∗ (Ψ∗ ) = Ψ∗ ⊗ Ψ∗ . Hence inv inv inv ∆∗ (Ψinv ∗ Ψ∗ ) = ∆∗ (Ψ∗ )∆∗ (Ψ∗ ) = Ψ∗ Ψ∗ ⊗ Ψ∗ Ψ∗ . But clearly X Ψinv ζ∗S(w)w. (6.7) ∗ Ψ∗ = 1 + w∈X∗ b

So Cor. 6.1.5 follows from Thm. 4.1.6 directly. Similar to Eq. (6.7) we also have X S Ψinv ζ
(w)w.
Ψ
= 1 +

(6.8)

w∈X∗ b

S S and MZ
Let MZw , MZ∗,w ,w be the Q-vector space generated by all weight w MZVs, ∗-symmetrized MZVs and -symmetrized MZVs, respectively.




Theorem 6.3.2. For all s = (s1 , . . . , sd ) ∈ Nd , the SMZVs S ζ
(s) ≡ ζ∗S(s) (mod ζ(2)), S S where ζ(2) means either ζ(2)MZ|s|−2 , or ζ(2)MZ∗,|s|−2 , or ζ(2)MZ
,|s|−2 . Consequently, in any statement modulo products of the MZVs we may write ζ S(s) for either one of the two symmetrized versions. Proof. According to Thm. 4.1.15 we have Ψ
(a, b) = ΛΨ∗ (a, b) where ∞ ∞ X X ζ(n) n  ζ(n) n  Λ = exp (−1)n b and Λinv = exp b . n n n=2 n=2 Since inv is an anti-automorphism inv Ψinv ΛΨ∗ = Ψinv
Ψ
= Ψinv ∗ Λ ∗ exp

∞ X ζ(2n) 2n  b Ψ∗ ≡ Ψinv ∗ Ψ∗ n n=2

S modulo ζ(2)MZ∗,|s|−2 . Similarly, we have ∞  X ζ(2n) 2n  inv inv −1 −1 inv Ψinv Ψ = Ψ (Λ ) Λ Ψ = Ψ exp − b Ψ
≡ Ψinv ∗
∗


Ψ
n n=2

S modulo ζ(2)MZ
,|s|−2 (or ζ(2)MZ|s|−2 by Prop. 6.1.3). Together with Eqs. (6.7) and (6.8) this completes the proof of the theorem.

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Remark 6.3.3. More detailed analysis reveals that ζ(2)MZ|s|−2 can be replaced by ζ(2)MZ|s|−2,≤dp(s)−1 where MZw,≤n is the subspace of MZw generated by the MZVs of depth at most n. Similar claims are true for the ∗- and -versions of the MZVs, too.




The next theorem shows that the SMZVs satisfy not only the stuffle relations but also some kind of the shuffle relations. Define τ (1) = 1 and τ (as1 −1 b . . . asd −1 b) = (−1)s1 +···+sr asd −1 b . . . as1 −1 b.

(6.9)

Note that τ and inv are the same as maps. But we should think inv as an anti-automorphism on (A1 , ∗, ∆∗ ) while τ is an anti-automorphism on (A1 , ).




Theorem 6.3.4. For all words u, w, w0 ∈ A1 , we have




S S (i) ζ
(u v) = ζ
(τ (u)v), S S (u τ (w)v), (ii) ζ
((wu) v) = ζ
S s−1 S (iii) ζ
((a bu) v) = (−1)s ζ
(u (as−1 bv)).













Proof. Taking u = ∅ and then w = u we see that (ii) implies (i). Decomposing w into strings of the type as−1 b we see that (iii) implies (ii). To prove (iii), we observe that  b (as−1 bu)


v − (−1) u
(a s

s−1

s−1  X bv) = (−1)i (as−1−i bu)


(a bv). i

i=0

(6.10) By Thm. 6.3.1 it suffices to show that the image of the above under E = Φ−1 bΦ vanishes. Now ∆
(E) = (Φ−1 ⊗ Φ−1 )(b ⊗ 1 + 1 ⊗ b)(Φ ⊗ Φ) = E ⊗ 1 + 1 ⊗ E. Thus E is a primitive element for ∆
so that we can regard it as a Lie element, namely, it acts on shuffle products like a “derivation”. By the definition, the counit 
satisfies 
[w] = 0 for all w 6= 1. Hence for any nonempty words u, v ∈ A1 , E[u


v] = E[u]
[v] + 
[u]E[v] = 0.

This completes the proof by Thm. 6.3.1 since none of the factors in the shuffle products on the right-hand side of Eq. (6.10) is empty word as the letter b appears in every factor.

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Symmetrized Multiple Zeta Values (SMZVs)

Besides the stuffle and the shuffle relations, MZVs satisfy quite a few other well-known relations such as the duality relations and the sum relations presented in Chap. 5. For the SMZVs, there is in fact another kind of duality which we call ϕ-dual. On the level of words it corresponds to the involution aϕ = a + b,

bϕ = −b.

(6.11)

Theorem 6.3.5. For all w ∈ A1
, we have ζ S(w) ≡ ζ S(wϕ )

where ζ (w) means either ζ
(w) or S

S

(mod ζ(2)),

ζ∗S(w).

Proof. Throughout this proof we write Φa,b = Φ(a, b). Set c = −(a + b). For any t ∈ Nd , we first decipher the coefficient Φa,c [at1 −1 b · · · batd −1 b].

Observe that the expansion of ar1 −1 (a + b) · · · ar` −1 (a + b) contains at1 −1 b · · · atd −1 b if and only if it has the form α1,1 · · · α1,t1 −1 (a + b) · · · αd,1 · · · αd,td −1 (a + b)

where αµ,ν = a or αµ,ν = a + b. Since the coefficient of ar1 −1 (a + b) · · · ar` −1 (a + b) in Φa,c is (−1)` Φa,b [ar1 −1 b · · · ar` −1 b] we get Φa,c [at1 −1 b · · · atd −1 b] X = (−1)d+β(b)} Φa,b [β1,1 · · · β1,t1 −1 b · · · β`,1 · · · βd,td −1 b], βµ,ν =a,b 1≤µ≤d,1≤ν···> kd 06=|kj |∈Z≤M ∀j=1,...,d


v (k1 , . . . , kd ) . k1s1 · · · kdsd

6.7. Prove Eq. (6.20) by applying  n  X 1 2 ··· i i + 1 ··· n (n) sh1 = , i 1 ··· i − 1 i + 1 ··· n i=1

to both sides of Eq. (6.19). 6.8. Show that the symmetry group Sn+1 is generated by the following three elements: Cn , Tn,0 ∈ Sn and Tn0 defined by Eq. (3.34).

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Chapter 7

Multiple Harmonic Sums (MHSs) and Alternating Version

A multiple harmonic sum (MHS) is a finite partial sum of the infinite series in Eq. (1.8) defining the MZV ζ(s1 , . . . , sd ) for s1 , . . . , sd ∈ N (note that s1 = 1 is allowed for partial sums). We have already encountered these numbers in previous chapters (see (3.18)). An alternating MHS (AMHS for abbreviation) is a finite partial sum of the infinite series defining a Euler sum, an alternating version of a MZV. All of these partial sums are rational numbers so it makes perfect sense to consider congruence properties modulo sufficiently large primes. This will be our first goal in this chapter. The second goal is to derive some identities using the binomial coefficients which will be useful to evaluate the MZVs and similar objects when the number of terms of the partial sums goes to infinity. 7.1 Definitions We start by defining a sort of double cover of the set N0 of nonnegative integers. Definition 7.1.1. Let D0 be the set of signed numbers N0 ∪ N0 where N0 := {k¯ : k ∈ N0 }.

¯ = k for all k ∈ N0 Define the absolute value function | · | on D0 by |k| = |k| ¯ and the sign function by sgn(k) = 1 and sgn(k) = −1 for all k ∈ N0 . Numbers in N0 are called signed numbers and numbers in N0 are called unsigned numbers. We make D a semi-group by defining a commutative and associative binary operation ⊕ (called O-plus) as follows: for all a, b ∈ D0  |a| + |b|, if sgn(a) 6= sgn(b); a ⊕ b := (7.1) |a| + |b|, if sgn(a) = sgn(b). Finally, we define D := D0 \ {0, ¯0}. 225

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Remark 7.1.2. We only use D in this chapter but we will need D0 to handle the q-analogs of the MHSs and the MH? Ss in Chap. 9. Definition 7.1.3. For any d ∈ N and s = (s1 , s2 , . . . , sd ) ∈ Dd , we define the alternating multiple harmonic sums by d X Y sgn(sj )kj , (7.2) Hn (s1 , s2 , . . . , sd ) = |s | kj j n≥k1 >k2 >...>kd ≥1 j=1 Hn? (s1 , s2 , . . . , sd ) =

X

d Y sgn(sj )kj |sj |

n≥k1 ≥k2 ≥...≥kd ≥1 j=1

kj

.

(7.3)

When s ∈ Nd these are called the multiple harmonic sums (MHSs) and the multiple harmonic star sums (MH? Ss), respectively. Conventionally, we Pd call dp(s) := d the depth and |s| := j=1 |sj | the weight. For convenience, we set Hn (s) = 0 if n < dp(s), Hn (∅) = Hn? (∅) = 1 for all n ≥ 0. For instance, the harmonic numbers and the alternating harmonic numbers have weight and depth both equal to 1: n n X X (−1)k 1 , Hn (¯1) = . Hn (1) = k k k=1

k=1

7.2 Stuffle Relations Throughout this section we let n be a fixed positive integer and consider only the AMHSs of the form Hn or Hn? . Recall the Hoffman algebra A1 = Q1 ⊕ Ab given in Definition 3.2.2 is a noncommutative polynomial algebra on generators zk := ak−1 b for all k ≥ 1. We further equip it with the stuffle product ∗ and regard it as a subalgebra of A∗ defined in Definition 3.2.7. To study both the MHSs and the AMHSs we need to generalize this algebra as sketched below. We omit the details because we will extend this framework even further to arbitrary level N in Chap. 13 by replacing ±1 by N th roots of unity (see Definition 13.3.2 there). Definition 7.2.1. For N ∈ N, we define the Hoffman–Racinet algebra of level N , denoted by AN , to be the (weight) graded noncommutative polynomial Q-algebra generated by N + 1 letters a, b0 , . . . , bN −1 . The number of letters in a word w is called its weight and the number of b0 , . . . , bN −1 , its depth. For every positive integer k, define zk := ak−1 b0 Let

A1N

and, if N = 2, zk¯ := ak−1 b1 .

(7.4)

be the subalgebra of AN generated by words not ending with a.

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Note that when N = 2, A12 is the subalgebra of A2 generated by words zk with k ∈ D. Remark 7.2.2. For future reference, when we deal with the Euler sums we will need to consider the subalgebra A02 of A2 generated by words beginning with a or b1 and ending with b0 or b1 . The words in A02 are called admissible words. To define the stuffle relations among AMHSs, a very simple example is already illuminating: Hn (¯ 1)Hn (2) = Hn (¯1, 2) + Hn (2, ¯1) + Hn (¯3). This follows immediately from the expansion X X X X · = + n≥j>0 n≥k>0

n≥j>k>0

n≥k>j>0

+

X

.

n≥j=k>0

Definition 7.2.3. For any k ∈ N, we define the toggling operator τzk as follows: ∀w ∈ A12 we set τzk (w) = w, and define τzk¯ (w) to be the word produced from w by exchanging zl and z¯l for all l. We define a multiplication ∗ on A12 by requiring that ∗ distribute over addition, that 1 ∗ w = w ∗ 1 = w for each word w ∈ A12 , and that, for all words u, v and special words x and y equaling either zk or zk¯ for some k,  

 xu ∗ yv = x τx τx (u) ∗ yv + y τy xu ∗ τy (v) 
 (7.5) + [x, y] τ[x,y] τx (u) ∗ τy (v) where [zk , zl ] = [zk¯ , z¯l ] = zk+l

and

[zk¯ , zl ] = [zk , z¯l ] = zk+l .

Using Eq. (7.5) we can further define a multiplication ∗? on A12 : simply replace ∗ by ∗? and substitute −[x, y] for +[x, y] in Eq. (7.5). The proof of the following stuffle relations is similar to that of MZV using induction on depth and is thus omitted here. Theorem 7.2.4. For any word w = zs1 . . . zsd ∈ A12 , define Hn (w) := Hn (s1 , . . . , sd ) and Hn? (w) := Hn? (s1 , . . . , sd ), and then linearly extend both over A12 (cf. Definition 3.2.2). Then for any words u, v ∈ A12 we have Hn (u)Hn (v) = Hn (u ∗ v),

Hn? (u)Hn? (v) = Hn? (u ∗? v).

(7.6)

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7.3 Quasi-Symmetric Functions with Signed Powers To derive more relations between the MHSs, the MH? Ss and the AMHSs we turn to the theory of quasi-symmetric functions. To treat the AMHSs with signed indices in D we have to allow the powers in these quasi-symmetric functions to be signed numbers. Definition 7.3.1. We denote by Z[x1 , . . . , xn ; D] the set of polynomials in x1 , . . . , xn with signed powers, namely, Z[x1 , . . . , xn ; D]   dn d1   X X ··· ce1 ,...,en xe11 · · · xenn d1 , . . . , dn ∈ N0 , ce1 ,...,en ∈ Z . =   en =dn e1 =d1 Pd Here, 0 = 0 and e=d means e runs through the set {d, . . . , 1, 0, 1, . . . , d}. 0 0 Furthermore, xej xej = xe⊕e for any j ≤ n and e, e0 ∈ D. Also we set i deg(xe11 · · · xenn ) = |e1 | + · · · + |en |. For positive integer n, we define a graded algebra homomorphism ψn : A12 −→ Z[x1 , . . . , xn ; D] X zs1 · · · zsd 7−→ xsk11 · · · xskdd , ∀d ≤ n, n≥k1 >k2 >···>kd ≥1

where we have set ψn (1) = 1, ψn (w) = 0 if the depth of w is greater than n, and the degree of every xi is equal to 1. It is easy to see we can arrange (ψn )n≥1 into a compatible system to obtain a homomorphism ψ from the Hoffman–Racinet algebra A12 to the algebra QSym2 of quasisymmetric functions which are defined below. Definition 7.3.2. Let x = (xj )j≥1 . An element of finite degree F (x) ∈ Z[[x; D]] is called a quasi-symmetric function if for any i1 > i2 > · · · > id , j1 > j2 > · · · > jd and any signed powers e1 , . . . , ed ∈ D the coefficients of the monomials xei11 · · · xeidd and xej11 · · · xejdd are the same. The set of all such quasi-symmetric functions is denoted by QSym2 . P ¯ For example, i>j≥1 xi x2j ∈ QSym2 but is not a symmetric function ¯ ¯ since the monomial x2 x21 appears in the sum but x1 x22 does not. We may choose an integral basis for QSym X 2 using Es = Es1 ,...,sd := xsk11 · · · xskdd , or k1 ≥k2 ≥···≥kd

Ms = Ms1 ,...,sd :=

X k1 >k2 >···>kd

xsk11 · · · xskdd = ψ(zs1 · · · zsd ),

(7.7)

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(see Exercise 7.5). This yields the following theorem which can be compared to [292, Thm. 2.2].

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Theorem 7.3.3. The map ψ provides an isomorphism (A12 , ∗) ∼ = QSym2 . We will see that the MHSs satisfy a different kind of duality from that of the MZVs. To derive it, we first define the power set of s to be the partial sum sequence: P (s) = (s1 , s1 +s2 , . . . , s1 +· · ·+sd−1 , w) as an (increasingly ordered) d-subset (i.e., subset of d elements) of (1, 2, . . . , w). We see that P provides a one-to-one correspondence between the compositions of weight w and depth d and the d-subsets of (1, 2, . . . , w). Let Rw (a1 , . . . , a` ) = (w + 1 − a` , . . . , w + 1 − a1 ). The v-dual of s is the composition of weight w corresponding to the complimentary subset of Rw P (s) in (1, 2, · · · , w). Namely,
s∨ = P −1 (1, 2, · · · , w) \ Rw P (s) . (7.8)

It’s easy to see that (s∨ )∨ = s so the v-dual is an involution. Another equivalent definition is the following (see Excercise 7.2). Definition 7.3.4. For positive integers r1 , . . . , r` , t1 , . . . , t` , let s = (r1 , {1}t1 −1 , r2 + 1, {1}t2 −1 , . . . , r` + 1, {1}t` −1 ). We define the v-dual of s by s∨ := ({1}r1 −1 , t1 + 1, {1}r2 −1 , t2 + 1, . . . , t`−1 + 1, {1}r` −1 , t` ).

(7.9)

From the definition we clearly have dp(s) + dp(s∨ ) = |s| + 1.

(7.10)

Remark 7.3.5. The v-dual can be easily explained using the conjugation on the ribbons (a kind of skew-Young diagrams). See Exercise 7.3. ϕ Recall that the ϕ-dual is defined on A11 by Eq. (6.11): xϕ 0 = x0 +x1 , x1 = −x1 . We transport it to QSym1 by Eq. (7.7). Namely, we set ϕ Msϕ := ψ(zϕ s1 · · · zsd ).

In the next theorem, the two kinds of dualities, v-dual and ϕ-dual, are related to each other in A11 . Theorem 7.3.6. Let s be a composition of positive integers. Then we have P (i) Msϕ = (−1)dp(s) ts Mt ; (ii) Esϕ = −Es∨ .

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Proof. (i) Suppose a word w ∈ A11 corresponds to the composition s. Then a replacement of some particular x0 by x1 in w corresponds to a splitting of s. Thus (i) follows. P (ii) Define Fs := ts Mt = (−1)dp(s) Msϕ by (i). Then, by the definition X ϕ X (−1)dp(t) Ft . (7.11) Esϕ = Mt = ts

ts

Dualizing [235, Cor. 3.16] we get S(Ft ) = (−1)|t| F← t ∨ where S is the antipode of QSym2 as defined in Thm. A.15. Thus, applying S to Eq. (7.11) we get X X X ∨ S(Esϕ ) = (−1)dp(t)+|t| F← (−1)dp(t ) F← (−1)dp(r) Fr t∨ = − t∨ = − ts

ts

←

r s ∨

because of Eq. (7.10). By M¨obius inversion formula we see that S(Esϕ ) = ←∨

−(−1)dp( s

)

M← s ∨ . Applying S again, we finally get ←∨

Esϕ = −(−1)dp( s

)

S(M← s ∨ ) = −Es∨

by Thm. A.15(i). Corollary 7.3.7. For any composition s of positive integers, X X − Es∨ = (−1)dp(r) Mt . rs

(7.12)

tr

Proof. By Thm. 7.3.6(ii) and the definition of Es we get X −Es∨ = Mrϕ . rs

So Eq. (7.12) follows from Thm. 7.3.6(i). We now apply the above results concerning quasi-symmetric functions to AMHSs. In order to do so, for any n ∈ N, we define the algebra homomorphism ρn : (A12 , ∗) → R such that ρn (Ms ) = Hn (s).

(7.13)

Theorem 7.3.8. For all s = (s1 , . . . , sd ) ∈ Dd and positive integers n, X X Hn (s) = (−1)dp(s)−dp(t) Hn? (t), Hn? (s) = Hn (t), (7.14) ts

ts

X

Hn? ( s ) = (−1)dp(s) ←

Fl

j=1

sj =s

(−1)r

r Y j=1

Hn (sj ),

(7.15)

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X

←

Hn ( s ) = (−1)dp(s)

(−1)r

Hn? (sj ).

j=1 sj =s

(7.16)

Moreover, for all s = (s1 , . . . , sd ) ∈ Nd X X Hn? (s∨ ) = − (−1)dp(r) Hn (t). rs

231

j=1

Fl

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r Y

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(7.17)

tr

Proof. The four equations (7.14), (7.15), (7.16), and (7.17) follow from the definition of Es , Thm. A.15(ii) and (iii), and Cor. 7.3.7, respectively, after we apply ρn . Definition 7.3.9. We will call the relations in Eqs. (7.15) and (7.16) the concatenation relations between the MHSs and the MH? Ss. 7.4 Symmetric Sums For a partition Π = {P1 , . . . , Pl } of the set {1, 2, . . . , d}, let c(Π) =

l Y

(]Pi − 1)!

and c˜(Π) = (−1)d−l

i=1

l Y

(]Pi − 1)!.

i=1

Let s = (s1 , . . . , sd ) ∈ Dd be a composition of depth d. Set Hn (s; Π) =

l Y

Hn (λi ),

Hn? (s; Π) =

i=1

l Y

Hn? (λi ),

i=1

where λi = ⊕j∈Pi sj . Further, let σ ∈ Sd act on s by permuting the subscripts. Theorem 7.4.1. For any d, n ∈ N and any compositions s ∈ Dd , X X
Hn? σ(s) = c˜(Π)Hn (s; Π), σ∈Sd

X σ∈Sd

(7.18)

Π: partitions of {1,2,...,d}


Hn σ(s) =

X

c(Π)Hn (s; Π).

(7.19)

Π: partitions of {1,2,...,d}

Proof. We prove the theorem under the assumption that each of the components s1 , . . . , sd of s satisfies that |sj | is a positive real numbers greater than 1 (of course we should first extend D to a double cover of R in the obvious way). Then by taking limit we can prove the theorem easily. Further we may assume all s1 , . . . , sd are pairwise distinct since again we can take limit.

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σ∈Sd

X
Hn σ(s) =

sgn(s1 )kσ(1) · · · sgn(sd )kσ(d)

X

|s |

|s |

1 d kσ(1) · · · kσ(d)

σ∈Sd k1 ≥···≥kd

.

Let Sd be the symmetry group of order d. Let Ξ(k) be the set of equivalence k

classes of {1, . . . , d} with i ∼ j if and only if ki = kj . Let O(k) = {σ ∈ k

Sd |σ(i) ∼ i ∀i = 1, . . . , d} be the isometry group of k = (k1 , . . . , kd ). Then sgn(s1 )kσ(1) · · · sgn(sd )kσ(d) |s |

|s |

1 d kσ(1) · · · kσ(d)

appears on the left-hand side of Eq. (7.18) exactly ]O(k) times. On the right-hand side it appears exactly in those terms corresponding to partitions Π refining Ξ(k), denoted by Π  Ξ(k), namely, Ξ(k) can be obtained from Π by combining some of its equivalence classes (see Thm. A.15). Hence, to prove Eq. (7.18) it suffices to prove that for all k = (k1 , . . . , kd ) X ]O(k) = c(Π). ΠΞ(k)

Indeed, c(Π) counts the permutations with the cycle type uniquely determined by Π while on the left-hand side every element of O(k) determines a unique cycle type prescribed by some Π  Ξ(k). We leave the proof of Eq. (7.19) as an exercise (see Exercise 7.6) for the interested reader. 7.5 Binomial Identities The binomial identities play very important roles in combinatorics and discrete mathematics in general. In this section we will consider some of these in the context of the MHSs and the MH? Ss. We start with an old formula known to Euler already. Proposition 7.5.1. For all positive integers n, we have   n X (−1)k n . Hn (1) = − k k k=1

Proof. Let f (x) =

n X xk k=1

k

.

(7.20)

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Multiple Harmonic Sums (MHSs) and Alternating Version

Then setting y = 1 − x   n n X X 1 − xn 1 − (1 − y)n n f 0 (x) = xk−1 = = =− (−1)k y k−1 . 1−x y k Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 06/12/16. For personal use only.

k=1

k=1

Integrating from 1 to x we get n X

  − x)k n . f (x) − f (1) = (−1) k k k=1 Pn So Eq. (7.20) follows by setting x = 0 as f (1) = k=1 k1 = Hn (1). k (1

We will see that more binomial identities come from Ap´ery’s famous series ∞ ∞ X 1 5 X (−1)k−1 ζ(2) = 3 , ζ(3) = (7.21)

, 2 k 2 2k k 3 2k k k k=1 k=1 which were used in his irrationality proof of ζ(2) and ζ(3) (see [19] or [556]). These have been generalized by Leshchiner [387] to all the Riemann zeta P∞ values and Dirichlet beta values β(n) = k=0 (−1)k /(2k+1)n . For example, he showed ∞  3 X (−1)m Hk−1 ({2}m ) 1  1 − 2m+1 ζ(2m + 2) =
2 2 k 2 2k k k=1 +2

m X ∞ X (−1)m−j Hk−1 ({2}m−j ) ,
k 2j+2 2k k j=1 k=1

∞

(−1)m−1 · ζ(2m + 3) =

5 X (−1)k Hk−1 ({2}m )
2 k 3 2k k k=1 m X ∞ X

+2

(−1)k−j Hk−1 ({2}m−j ) .
k 2j+3 2k k j=1 k=1

In the rest of this section we state a few families of the binomial identities for the MH? Ss similar to Prop. 7.5.1. We will apply some of these results to a few congruences of the MHSs and the MH? Ss in Chap. 8 and then to some identities of the multiple zeta star values in Chap. 10. Theorem 7.5.2. Let n, c ∈ N and a, b ∈ N0 . If c ≥ 3 then
n X (−1)k−1 nk ? a b Hn ({2} , c, {2} ) =2
k 2a+2b+c n+k k k=1 n X X Hk−1 (x, i + 2b)(−1)k−1 +4 2dp(x)
k 2a+j n+k k i,j,x k=1

n k

(7.22)


,

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where in the second sum i ≥ 1, j ≥ 2 and x ∈ Nr runs through all possible compositions of positive integers (or the empty set when r = 0) satisfying the restriction i + j + |x| = c. Remark 7.5.3. If we take c = 3 then we obtain so-called Two-three formula. Here “Two” is capitalized to denote a 2-string. Theorem 7.5.4. For all n, c ∈ N and a, b ∈ N0 , we have
n X (−1)k−1 nk ? a b Hn ({1} , c, {1} ) = k a+b+c k=1

+

X

n X Hk−1 (b + i, s)(−1)k−1

i+j+|s|=c k=1 i≥1,j≥1,|s|≥0

k a+j

n k

(7.23)


.

The following formulas are special cases of so-called Two-one formula. Theorem 7.5.5. Let a ∈ N0 and b ∈ N. Then for any n ∈ N, we have
n n X ? a k (7.24) Hn ({2} , 1) = 2
, k 1+2a n+k k k=1

n n X X (−1)k nk Hk−1 (2b) nk ? a b Hn ({2} , 1, {2} ) = −2
−4
. (7.25) k 1+2a+2b n+k k 1+2a n+k k k k=1 k=1 The proofs of the above theorems are not given here since we will generalize them to the case with argument strings having arbitrary times of repetitions of certain patterns (for example, ({2}a1 , c1 , . . . , {2}ar , cr ) with r ≥ 1) in Chap. 9, and moreover, generalize further to their q-analogs, which will be used in Chap. 10 where the multiple zeta star values are treated. 7.6 Historical Notes Even before the appearance of renewed interest in MZVs in the early 1990s, physicists already started to study the MHSs and the AMHSs. In [165], Devoto and Duke listed a lot of concrete computations involving such sums. Later on, Bl¨ umlein and Kurth [58,61] and Vermaseren [560] discovered that they can be realized as the general basis for the Mellin transforms of all individual functions of the momentum fraction x emerging in the quantities of massless quantum electrodynamics (QED) and quantum chromodynamics (QCD), and as Mellin transforms and the inverse Mellin transforms for some functions that are encountered in Feynman diagram calculations,

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235

and some other similar functions with significant physical meanings. Incidentally, our binary operation ⊕ for the signed numbers is denoted by the ampersand & by Vermaseren in [560]. Later, it is also noticed that the MHS structure and its cyclotomic extensions are closely linked to high energy physics (see [3, 59, 60]) and the relations between these sums can be used to simplify certain results of higher-order calculations in QED and QCD. The first systematic study of the MHSs from the number theoretical perspective was initiated by the author in the arXiv paper [609] in early 2003, and independently by Hoffman in the arXiv paper [292] in early 2004. In particular, the approach using algebras of words and quasi-symmetric functions to study MZVs was carried over to this new setting by Hoffman. Note that the star-version H ? is also denoted by S in [292] but it seems to be more appropriate to use H ? in this book due to its close connection with multiple zeta star values to be considered in Chap. 10. Quasi-symmetric functions are generalizations of symmetric functions introduced by Gessel and Stanley in early 1980s to investigate the combinatorics of P -partitions and the counting of permutations with given descent sets, see [236, 237, 525]. They can be regarded as the dual algebra over the integers of the Leibniz-Hopf algebra (see [275]). In [289], Hoffman first treated MZVs from the point of view of two different product structures and later he extended these to the Hopf algebra on both the MZVs and the MHSs in [290, 293]. Our proof of Thm. 7.4.1 is adopted from that of Thms. 2.1 and 2.2 of [288]. Note that [288, Thm. 2.2] was stated as Prop. 9.4 in [368] whose proof is different from that contained in [288] but also works here in the alternating setting. We saw a few different types of sum relations among the MZVs in Chap. 5. Theorem 7.4.1 is an analog of such formulas in the AMHS setting, which is a generalization of Thms. 2.1 and 2.2 of [288] concerning only the MHSs. Using binomial identities, Hessami Pilehroods and Tauraso [278] discovered many congruences for MHSs and MH? Ss appearing on the lefthand side of Eqs. (7.23)-(7.25) that will be discussed in Sec. 8.6.4. Theorem A.15 generalizes [294, Thm. 6.2]. Known already to Euler [211], Prop. 7.5.1 was the starting point from where Wolstenholme (see [578]), in 1862, began to prove the famous result now bearing his name (see Thm. 8.2.1). Our proof of Thm. 7.3.6 follows that of [294, Thm. 6.3]. Finally, we note that the detailed proofs of Thms. 7.5.2 to 7.5.5 can be found in [278] which motivated a complete proof of the Two-one formula

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by the author in [628].

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Exercises 7.1. By Definition 7.8 find the v-duals s∨ if (i) s = (1, 3, 2) (ii) s = (4, 2, 1, 3). Then verify them using Definition 7.3.4. 7.2. Show that Eq. (7.8) and Eq. (7.9) are equivalent. 7.3. In this exercise, we re-interprete the v-dual using the following operation on the ribbons, i.e., skew-Young diagrams. For a composition s = (s1 , . . . , sd ), the ribbon Rs is defined to be the skew-Young diagram of d rows whose jth row starts below the last box of (j − 1)st row and has exactly sj boxes. Recall that the conjugate of a (skew-)Young diagram is the mirror image about the diagonal line going from the south-west corner to north-east. For example, the following two diagrams give the ribbon R1,3,2 and its conjugate:

Show that in general the ribbon Rs∨ is exactly the conjugate of the ribbon ∨ R← s . So (2, 3, 1) = (1, 2, 1, 2). 7.4. Prove Thm. 7.2.4. 7.5. Check that the set {Mt1 ,...,td : d ∈ N, (t1 , . . . , td ) ∈ Dd } forms an integral basis for QSym2 . 7.6. Prove Eq. (7.19) by modifying the argument for the proof of Eq. (7.18).

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Chapter 8

Finite Multiple Zeta Values and Finite Euler Sums

A general philosophy in studying congruences is to consider so-called van Hamme type congruences. One starts with an intriguing infinite series whose terms are given by rational numbers with some pattern and then, for suitable primes p, looks at the (p − 1)-st partial sum modulo p, or even modulo higher powers of p which leads to super congruences. A classical result in this spirit is a variant of Wolstenholme’s Theorem dating back to the mid nineteenth century: for all prime p ≥ 5 we have p−1 X 1 ≡0 k2

p−1 X 1 ≡0 k

(mod p),

(mod p2 ).

(8.1)

k=1

k=1

These can be proved rather easily as follows. First, the map f (x) = x−2 is an automorphism of (Z/pZ)× so the first congruence is obvious. For the second, consider the equations in the ring Zp of p-adic integers:   p−1 p−1  p−1  X X 1 X 1 1 1 1  p −1 2 = + = − · 1− k k p−k k k k k=1

k=1

k=1

≡ −p

p−1 X 1 ≡0 k2

(mod p2 )

k=1

by the first congruence of (8.1). These were improved to super congruences later (see Sec. 8.7). In the proof of these congruences one finds that Bernoulli numbers play the key roles by virtue of the Seki–Bernoulli formula1 :  n−1 d  X X d + 1 Br d+1−r d j = n , ∀n, d ≥ 1. (8.2) r d+1 r=0 j=1 1 See [22, p. 3]. This formula is also called Faulhaber’s formula in the literature, see [6, § 23.1.4-7].

237

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This chapter gives a brief introduction to the van Hamme type (super) congruences for the (alternating) multiple harmonic (star) sums. A general phenomenon is that in order to study the weight w MHSs (or MH? S) modulo pk+1 one must first study those objects of weight w + k modulo p. A few important ideas have been used to investigate this structure by a number of mathematicians. Generally speaking, they all use some kind of combinations of the following four approaches: (1) rearrange the sum, (2) apply the stuffle, shuffle, reversal, concatenation, and duality relations, (3) change to the sums of positive powers and then use the Bernoulli numbers/polynomials, and (4) exploit the binomial identities. Throughout this chapter we will follow the above approaches closely. 8.1 Definitions Let P be the set of primes and let ` be a positive integer. Define M Y ` A` := (Z/p Z) (Z/p` Z) p∈P

(8.3)

p∈P

with componentwise addition and multiplication. Clearly the elements of A` are represented by (ap )p∈P and two elements (ap ) and (bp ) represent the same element in A` if ap = bp for all but finitely many primes p. For every nonzero r ∈ Q, by the fundamental Theorem of Arithmetic Y r= pordp (r) . p∈P

It is straightforward to see that A` is a Q-algebra after embedding Q in A` “diagonally” using the following map: ι` : Q −→

A`


r −→ ιp,` (r) p∈P

(8.4)

where ι` (0) = (0)p∈P and for all nonzero r ∈ Q  0, if ordp (r) < 0; ιp,` (r) := ` r (mod p ), if ordp (r) ≥ 0. Proposition 8.1.1. The map ι` is a monomorphism of algebras. Namely, ι` gives an embedding of Q into A` as an algebra.

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Proof. Suppose r, s ∈ Q with ι` (r) = ι` (s). Note that ordp (r) < 0 or ordp (s) < 0 for only finitely many primes p. Thus ι` (r) = ι` (s) implies that r − s is divisible by infinitely many primes which immediately yields r = s. This shows ι` is injective. Since we clearly have ι` (r)ι` (s) = ι` (rs) for all r, s ∈ Q the injection ι` is an algebra homomorphism. Remark 8.1.2. It is easy to see that the cardinality of A` is ℵ1 , namely, the same as that of the real numbers. See Exercise 8.1. It is not too hard to construct an embedding of R into A` as sets. But we don’t know whether there is an embedding of R as an algebra. For convenience, we abuse our notation by writing ap for (ap )p∈P if no confusion arises. In particular, whenever p appears in an equation in A` (` ≥ 2) it means the element (p)p∈P. Also, by writing (ap )p≥k ∈ A` we mean the element (ap )p∈P with ap = 0 for all p < k. Definition 8.1.3. We define a number a ∈ A` to be algebraic over Q if there is a nontrivial polynomial f (t) ∈ Q[t] such that f (a) = 0. Nonalgebraic numbers of A` over Q are called transcendental. For finitely many numbers a1 , . . . , an ∈ A` , we say they are algebraically independent over Q if for any nontrivial polynomial f (t1 , . . . , tn ) ∈ Q[t1 , . . . , tn ] we have f (a1 , . . . , an ) 6= 0. We call the elements in an infinite subset S of A` algebraically independent over Q if any finitely many elements of S are. Definition 8.1.4. For any  nonnegative integer k, we define the A1 B Bp−k ≡ p−k (mod p) . For all k ≥ 2, Bernoulli numbers βk := − p−k k p>k

define the kth (A1 -)Fermat quotient qk :=

 k p−1 − 1 p

(mod p)

 p>k

.

We have β2k = 0 for all k ≥ 1 since all Bernoulli-numbers B2j+1 = 0 when j ≥ 1. As for the A1 -Fermat quotient q2 , according to [348], we know that for all primes less than 1.25 × 1015 the super congruence 2p−1 ≡ 1 (mod p2 ) is satisfied by only two primes 1093 and 3511 which are called Wieferich primes. It is also known that qk 6= 0 (k ≥ 2) in A1 under abcconjecture (see [514]). Moreover, note that we have qk` = qk + q` for all k, ` and q1 = 0. So qk is an A1 -analog of the logarithm value log k. By considering the mysterious relation between the Euler sums and their finite versions (see Sec. 8.6.3) we come to the following hypothesis.

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Conjecture 8.1.5. Put β1 = 1 by abuse of notation. Suppose n1 , . . . , nr ∈ N such that log n1 , . . . , log nr are Q-linearly independent. Then the numbers in the set ∞ n o [ β2k+1 qn1 , . . . , β2k+1 qnr k=0

are algebraically independent. One should compare this with the following Conjecture 8.1.6. Put ζ(1) = 1 by abuse of notation. Suppose n1 , . . . , nr ∈ N such that log n1 , . . . , log nr are Q-linearly independent. Then the numbers in the set ∞ n o [ ζ(2k + 1) log n1 , . . . , ζ(2k + 1) log nr k=0

are algebraically independent. By the same token, every object similarly defined for each prime can be put into A1 , for examples, Wilson quotients (p−1)!+1 p p-adic zeta values ζp (k) Fp−( p5 ) /p: the Fibonacci quotient (OEIS A092330) Up−( p2 ) (2, −1)/p: the Pell quotient (OEIS A000129) Up−( p3 ) (4, 1)/p: a quotient related to the Lucas sequence 1, 4, 15, 56, 209, . . . (OEIS A001353) • Up−( p6 ) (10, 1)/p: a quotient related to the Lucas sequence 1, 10, 99, 980, 9701, . . . (OEIS A004189)

• • • • •

Definition 8.1.7. For s = (s1 , . . . , sd ) ∈ Dd (see Definition 7.1.1), we define X sgn(s1 )k1 · · · sgn(sd )kd ζA` (s) := ∈ A` , |s | |s | k1 1 · · · kd d p>k1 >···>kd ≥1 X sgn(s1 )k1 · · · sgn(sd )kd ? ζA (s) := ∈ A` . ` |s | |s | k1 1 · · · kd d p>k1 ≥···≥kn ≥1 These are called finite Euler (star) sums, abbreviated as FES. If all the sj ’s are positive then they are called finite multiple zeta (star) values, abbreviated as FMZV. The number ` is called the superbity.

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8.2 Reversal and Concatenation Relations

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The famous Wolstenholme’s Theorem stated in Eq. (8.1) can be rephrased in the following way. Theorem 8.2.1. For any prime p > 3, we have Hp−1 (1) ≡ 0

(mod p2 )

Hp−1 (2) ≡ 0

and

(mod p).

In order to extend these results to the general MHSs, we start in this section with some simple properties of the MHSs, namely, the reversal and stuffle relations. ← For any s = (s1 , . . . , sd ) ∈ Dd , denote its reversal by s = (sd , . . . , s1 ) Qd and set sgn(s) = j=1 sgn(sj ). The following results are called reversal relations. Theorem 8.2.2. Let s = (s1 , . . . , sd ) ∈ Dd . Then ←

ζA1 ( s ) = (−1)|s| sgn(s)ζA1 (s),

←

? ? ζA ( s ) = (−1)|s| sgn(s)ζA (s). 1 1

(8.5)

Proof. Let p be a prime at least 3. The theorem follows easily from the index substitution k → p − k and the following congruence which is valid for all s ∈ D: sgn(s)k  p −|s| sgn(s)p−k |s| = (−1) sgn(s) 1 − k (p − k)|s| k |s|   2   |s|p |s| + 1 p 1 |s| k+1 + |s|+1 + (mod p3 ). ≡ (−1) sgn(s) |s| |s|+2 2 k k k (8.6) The rest is straightforward and is left to the interested reader. Sometimes this idea can be applied to obtain super congruences, namely, with moduli given by higher powers of p. Theorem 8.2.3. Let s be any positive integer. Then ζA1 (2s) = 0,

and

ζA2 (2s + 1) = 0.

(8.7)

Proof. Suppose the multiplicative group (Z/pZ)× has a generator gp . Then in A1 Hp−1 (s) =

p−2 X k=0

s(p−1)

gpsk =

1 − gp 1 − gpn

= 0.

This argument fails if s = p − 1 in which case Hp−1 (s) = −1 in Z/pZ. But in A1 this is negligible since s is fixed and we can neglect any number of

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finitely many primes. Therefore ζA1 (s) = 0 for every positive integer s. Now if s = 2m + 1 is odd then by taking reversal we see that in Zp  p−1  X 1 1 + 2Hp−1 (2m + 1) = k 2m+1 (p − k)2m+1 k=1

≡

p−1 X (2m + 1)p k=1

k 2m+2

≡0

(mod p2 ),

because of the expansion in Eq. (8.6). This completes the proof of the theorem. We now lift Thm. 8.2.2 to superbity two by exactly the same idea. Theorem 8.2.4. Let s = (s1 , . . . , sd ) ∈ Dd . Let ei = (0, . . . , 0, 1, 0, . . . , 0) where 1 appears at the ith component. Then d X ← (−1)|s| sgn(s)ζA2 ( s ) = ζA2 (s) + p |si |ζA2 (s ⊕ ei ), (8.8) i=1

←

? ? (−1)|s| sgn(s)ζA ( s ) = ζA (s) + p 2 2

d X i=1

? |si |ζA (s ⊕ ei ), 2

(8.9)

where the binary operation ⊕ is carried out componentwise by Eq. (7.1). Proof. Take reversal and then apply Eq. (8.6). To find a result corresponding to Thm. 8.2.3 for signed numbers the Euler polynomials and the Euler numbers are indispensable tools. Recall that the Euler polynomials En (x) are defined by the generating function ∞ X 2etx tn = En (x) . t e + 1 n=0 n! Lemma 8.2.5. Let n ∈ N0 . Then we have d−1 n    X X 1 n i n d−1 (−1) i = (−1) En (d) + En (0) = Fn,d,a dn−a , (8.10) 2 a a=0 i=1 where Fn,d,a

 d−1  (−1) Ea (0)/2, = (1 − (−1)d )En (0)/2,  −(1 + (−1)d )/2,

if a < n; if a = n > 0; if a = n = 0.

Moreover, E0 (0) = 1 and for all a ∈ N 1  2a+1  2 Ea (0) = Ba+1 − Ba+1 = (1 − 2a+1 )Ba+1 . a+1 2 a+1

(8.11)

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Proof. Consider the generating function ! ∞ d−1 d−1 X X tn X (−et )d − 1 i n = −1 (−1) i (−1)i eti = n! i=1 −et − 1 n=0 i=1

(−1)d−1 edt + 1 −1 et + 1 ∞  tn 1 X = (−1)d−1 En (d) + En (0) − 1. 2 n=0 n! =

(8.12)

Now Eq. (8.10) follows from the notorious equation (see, for e.g., [6, § 23.1.7]) n   X n En (x) = Ea (0)xn−a (8.13) a a=0 for all n > 0. For Eq. (8.11), see, for e.g., § 23.1.20 on p. 805 of loc. cit. Remark 8.2.6. The classical Euler numbers Ek is defined by ∞ X 2 tk = Ek . t −t e +e k! k=0

They are related to Ek (0) by the formula  m−k m   X m Ek 1 Em (0) = − . 2 k 2k k=0

See § 23.1.7 on p. 805 of loc. cit. Theorem 8.2.7. Let s ∈ N. Then  −2q2 , ζA1 (¯ s) = −2(1 − 21−s )βs , ζA2 (¯ s) = s(1 − 2−s )pβs+1

if s = 1; if s > 2 is odd;

(8.14)

if s is even.

(8.15)

Proof. Taking d = p and n = p(p − 1) − s in Lemma 8.2.5 we see that Hp−1 (s) ≡ Fp(p−1)−s,p,p(p−1) + p(p(p − 1) − s)Fp(p−1)−s,p,p(p−1)−1−s 1 ≡ Ep(p−1)−s (0) − psEp(p−1)−1−s (0) (mod p2 ), 2 since all the coefficients in Eq. (8.10) are p-integral by Eq. (8.11). Note that Bm is not p-integral if and only if p − 1 divides m > 0. Then the theorem follows directly from Eq. (8.11) and the Kummer congruences Bp(p−1)−s Bp−1−s ≡ , p(p − 1) − s p−1−s

Bp(p−1)−s+1 Bp−s ≡ (mod p). p(p − 1) − s + 1 p−s

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Using the theory of quasi-symmetric functions one can get: Theorem 8.2.8. For all s = (s1 , . . . , sd ) ∈ Dd and positive integer `, X X ? ? (−1)dp(s)−dp(t) ζA (t). (8.16) ζA` (t), ζA` (s) = ζA (s) = ` ` ts

ts

When ` = 1 we have ←

X

? (−1)dp(s) ζA (s) = 1 Fr

j=1

←

sj =s

X

(−1)dp(s) ζA1 ( s ) = Fr

(−1)r

(−1)r

j=1 sj =s

r Y

ζA1 (sj ),

(8.17)

? ζA (sj ). 1

(8.18)

j=1 r Y j=1

Proof. These follow immediately from Thm. 7.3.8. Definition 8.2.9. We will call the relations in (8.17) and (8.18) the concatenation relations between FMZVs and FESs. Example 8.2.10. By (8.16) and the concatenation relation (8.17) we have ? ζA1 (1, ¯ 2) + ζA1 (¯ 3) = ζA (1, ¯2) = ζA1 (¯2)ζA1 (1) − ζA1 (¯2, 1). 1

Thus we get ζA1 (¯ 3) = −2ζA1 (¯2, 1) since ζA1 (1) = 0 and ζA1 (1, ¯2) = ζA1 (¯2, 1) by the reversal relation (8.5). 8.3 Stuffle Relations We may obtain more congruences by combining the reversal relations with the stuffle relations in Thm. 7.2.4. Definition 8.3.1. To find as many Q-linear relations as possible in weight w we may choose all the known relations in weight k < w, multiply them by ζA` (s) for all compositions s of signed numbers of weight w − k, and then expand all the products using the stuffle relations (7.5). Such Qlinear relations among the FESs of the same weight produced in this way are called linear stuffle relations of the FESs. For example, if k and l have the same parity and same sign (or have different parities and different signs) then 0 = ζA1 (k)ζA1 (l) = ζA1 (k, l) + ζA1 (l, k) + ζA1 (k ⊕ l) = 2ζA1 (k, l) by Eqs. (8.7), (7.6) and (8.5). When k, l > 0 this is a special case of the more general Thm. 8.5.12. An application of the stuffle relation yields:

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Theorem 8.3.2. Let s and t be two positive integers.

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(i) If s + t is odd then ? ζA (s, t) = ζA2 (s, t). 2

(8.19)

? (s, t, s) = ζA2 (s, t, s). ζA 2

(8.20)

? ζA (s, t, s) = ζA3 (s, t, s). 3

(8.21)

(ii) If t is even. Then

(ii) If t is odd. Then

Proof. Part (i) follows easily from Thm. 8.2.3 since ? (s, t) = ζA2 (s, t) + ζA2 (s + t). ζA 2

For part (ii), by (8.16) and the stuffle relation we get ? ζA (s, t, s) = ζA2 (s, t, s) + ζA2 (s + t, s) + ζA2 (s, t + s) + ζA2 (2s + t) 2

= ζA2 (s, t, s) + ζA2 (s + t)ζA2 (s) = ζA2 (s, t, s) by Thm. 8.2.3, as desired. If t is odd then s and s + t have different parity. Thus ζA3 (s + t)ζA3 (s) = 0 in A3 . This proves part (iii). 8.4 Shuffle Relations It is not hard to see that the Euler sums can be expressed by iterated integrals, just like MZVs. Suppose sj ∈ D and sgn(sj ) = ηj ∈ {1, −1} for all j = 1, . . . , d. Then   |sd |−1   Z 1  |s1 |−1  dt dt dt dt ... , ζ(s1 , . . . , sn ) = t ξ1 − t t ξd − t 0 (8.22) Qi where ξi = j=1 ηi for i = 1, . . . , d. Recall from Definition 7.2.1 that A12 is the algebra generated by zs , s ∈ D. Write zs = y|s|,sgn(s) . Then we define tog, tog

where ηi =

Qi

−1

:

A12

→

A12

by

tog(ys1 ,ξ1 · · · ysd ,ξd ) := ys1 ,η1 · · · ysd ,ηd ,

j=1 ξj

(8.23)

(8.24)

for i = 1, . . . , d and

tog−1 (ys1 ,η1 · · · ysd ,ηd ) := ys1 ,ξ1 · · · ysd ,ξd ,

−1 where ξj = ηj−1 ηj for j = 1, . . . , d if we set η0 = 1.

(8.25)

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For all s1 , . . . , sd ∈ D, we define the one-variable multiple polylog X

Ls1 ,...,sd (z) =

z k1

|sj |

j=1

k1 >k2 >...>kd ≥1

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d Y sgn(sj )kj

kj

,

0 < |z| < 1,

which generalizes Eq. (3.20) to the larger index set D. It satisfies   |sd |−1   Z z  |s1 |−1  dt dt dt dt ··· , (8.26) Ls1 ,...,sd (z) = t ξ1 − t t ξd − t 0

where ξi = sgn(si−1 ) sgn(si ), i = 1, . . . , d. Here we have set sgn(s0 ) = 1. Recall from Eq. (8.23) that zs = y|s|,sgn(s) for all s ∈ D and zs = zs1 . . . zsd ∈ A12 for all s = (s1 , . . . , sd ) ∈ Dd . Further, define the map L(−; t) : (A12 , ∗) → R[[t]] by setting L(zs ; t) = Ls (t) for all s ∈ Dd and the map L
(−; t) : (A12 , ) → R[[t]] by setting L
(w; t) = L(tog(w); t).




Proposition 8.4.1. The map L
(−; t) : (A12 , homomorphism.


) → R[[t]] is an algebra

Proof. This follows from (8.26) and the shuffle relation satisfied by the iterated integrals. Example 8.4.2. Suppose s = (¯2, 1). Then zs = y2,−1 y1,1 and 2 Z 1  −dx dx L(y2,−1 y1,1 ; t) = = L¯2,1 (t). x 1+x 0 Set ζA1 ,
= ζA1 ,∗ ◦ tog : A12 → A1 . Although ζA1 ,
is not an algebra homomorphism from (A12 , ) we will see in the next theorem that it does provide a kind of shuffle relation. It is the FES analog of Thm. 6.3.4 for the SMZVs. Recall that τ : A11 → A11 is defined by (6.9).




Theorem 8.4.3. For all words w, u ∈ A11 and v ∈ A12 , we have




(i) ζA1 ,
(u v) = ζA1 ,
(τ (u)v), (ii) ζA1 ,
((wu) v) = ζA1 ,
(u τ (w)v), (iii) For all s ∈ N, ζA1 ,
((xs−1 x1 u) v) = (−1)s ζA1 ,
(u 0











(x

s−1 x1 v)). 0

Proof. Taking u = ∅ and then setting w = u we see that (ii) implies (i). Decomposing w into strings of the type xs−1 x1 we see that (iii) implies (ii). 0 So we only need to prove (iii). For simplicity, let a = x0 and b = x1 in the rest of this chapter. Let sj ∈ D with sgn(sj ) = ηj for j = 1, . . . , d. Let u = zs and tog(u) = yt1 ,ξ1 . . . ytd ,ξd ∈ Dd . Then clearly tog(bu) = b tog(u).

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For any prime p > 2, the coefficient of tp in L
(bu; t) is given by

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  1 Coeff tp L
(bu; t) = p

X p>k1 >···>kd

ξ1k1 · · · ξdkd 1 td = Hp−1 (tog(u)). t1 p · · · k k d >0 1

Observe that  b (as−1 bu)


v − (−1) u
(a s

s−1

s−1  X bv) = (−1)i (as−1−i bu)


(a bv). i

i=0

By first applying L
(−; t) to the above and then extracting the coefficients of tp from both sides we get

 1 Hp−1 ◦ tog (as−1 bu) v − (−1)s Hp−1 ◦ tog u (as−1 bv) p s−1 X   = (−1)i Coeff tp L
(as−1−i bu; t) L
(ai bv; t)







i=0

=

s−1 X i=0

(−1)i

p−1 X j=1

    Coeff tj L
(as−1−i bu; t) · Coeff tp−j L
(ai bv; t)

by Prop. 8.4.1. Now the last sum is p-integral since p − j < p and j < p and therefore we get Hp−1 ◦ tog((as−1 bu)


v) ≡ (−1) H s

p−1

◦ tog(u


(a

s−1

bv))

(mod p)

which completes the proof of (iii). Definition 8.4.4. A relation produced by Thm. 8.4.3(i) is called a linear shuffle relation of the FESs. For each weight w ≥ 2, by the double shuffle relations of the FESs of weight w we mean all the linear shuffle relations of weight w and all the linear stuffle relations of w defined in Definition 8.3.1. Restricting to FMZVs, we obtain the linear shuffle relations and the double shuffle relations of the FMZVs. 8.5 Finite Multiple Zeta Values (FMZVs) We now apply the results from the previous sections to the FMZVs. 8.5.1

Homogeneous FMZVs

The following homogeneous generalization of Thm. 8.2.3 will be very handy, even though we will extend it further.

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Theorem 8.5.1. Let s and d be two positive integers. Then

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ζA1 ({s}d ) = 0 d

ζA2 ({s} ) = 0

if the weight ds is even;

(8.27)

if the weight ds is odd.

(8.28)

Proof. We may proceed by induction on d. The case d = 1 is just Thm. 8.2.3. Assume the theorem holds for d ≥ 2. Then 0 = ζA2 (s)ζA2 ({s}d−1 ) = dζA2 ({s}d ) +

X
1 ζA2 σ(2s, {s}d−2 ) . (d − 2)! σ∈Sd

So the theorem follows from Thm. 7.4.1 after counting the weight.

Our goal in this section is to prove an improvement of Thm. 8.5.1. First, we need some preliminary results. Lemma 8.5.2. Let p > 3 be a prime and let r be a positive integer such that r ≤ p − 3. Then  p−1 − r(r+1) Bp−r−2 p2 (mod p3 ) X 1 2(r+2) ≡ Hp−1 (r) =  r B kr (mod p2 ) k=1 r+1 p−r−1 p

if r is odd; if r is even.

The proof is standard so we leave it as Exercise 8.2. Lemma 8.5.3. Let r, α1 , · · · , αn be positive integers, r = α1 + α2 + · · · + αn ≤ p − 3, then X lα1 1≤l1 ,··· ,ln ≤p−1 1 li 6=lj

 (−1)n n!r(r+1)  2n(r+2) Bp−r−2 p2 (mod p3 )

1 ≡  (−1)n−1 n!r · · · lnαn B n(r+1)

p−r−1 p

(mod p2 )

if r is odd; if r is even.

Proof. If n = 1, the congruence is obvious by Lemma 8.5.2. We assume that the formula holds when the number of variables is less than n. Then

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by induction X

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l α1 1≤l1 ,··· ,ln

···>jd ≥1

|s |

j1 1 · · · jd d

jdp−a−k

which is exactly the right-hand side of Eq. (8.48). Theorem 8.6.2. Let a, d ∈ N, s = (s1 , . . . , sd ) ∈ Dd and s0 = (s1 , . . . , sd−1 ). For any prime p ≥ a + 2, write H(−) = Hp−1 (−). Then  (1 − 2p−a )Bp−a  H(s) − H(s0 , sd ) p−a p−2−a X p − 1 − a (1 − 2k+1 )Bk+1
− H s0 , (k + a) ⊕ sd (mod p), k k+1

H(s, a) ≡

k=0

where sd = t if sd = t ∈ N. Proof. By the definition and Lemma 8.2.5 H(s, a) ≡ p−2−a X 

X p>j1 >···>jd

jd −1 sgn(s1 )j1 · · · sgn(sd )jd X (−1)j j p−1−a |s | |s | j1 1 · · · jd d j=1

 X p − 1 − a Ek (0) sgn(s1 )j1 · · · sgn(sd−1 )jd−1 sgn(sd )jd ≡ |sd−1 | |sd |+k+a |s | k 2 p>j >···>j (−1)jd −1 j1 1 · · · jd−1 jd k=0 1 d   j j X Ep−1−a (0) sgn(s1 ) 1 · · · sgn(sd ) d + 1 − (−1)jd |s1 | |sd | 2 j · · · j p>j1 >···>jd 1 d 0   a −1  X a0 E (0)   Ea0 (0)  k ≡ H(s) − H(s0 , sd ) − H s0 , (k + a) ⊕ sd , 2 k 2 k=0

where a0 = p − 1 − a. The theorem follows from Eq. (8.11) easily. 8.6.2

FESs of Small Depths

In this section we study FESs in depth 2 and 3. All but one case are given by very concise values involving Bernoulli numbers or Euler numbers (which are themselves related by the identity Eq. (8.11)).

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Lemma 8.6.3. Let a, b ∈ N and prime p ≥ w + 3. Let w = a + b. Then (
p−w (−1)a 2 w −1 wa Bp−w (mod p), if w is odd; Hp−1 (¯ a, ¯b) ≡ (2a −2)(2b −2) Bp−a Bp−b (mod p), if w is even. 2w−1 ab Proof. Set H = Hp−1 . By the change of variable kj → p − kj for odd indices kj (j = 1, 2), we have H(¯ a, ¯b) + H(a, b) X ≡2

1

≡

X

+2

ka kb p>k1 >k2 >0,2|k1 ,2|k2 1 2

p>k1 >k2 >0,2-k1 ,2-k2

 2  H p−1 (a, b) + (−1)w H p−1 (b, a) w 2 2 2

1 k1a k2b

(mod p)

(mod p).

Similarly, H(a, b) ≡ H p−1 (a, b)+(−1)w H p−1 (b, a)+(−1)b H p−1 (a)H p−1 (b) 2

2

2

2

(mod p).

Thus H p−1 (a, b)+(−1)w H p−1 (b, a) ≡ H(a, b)−(−1)b H p−1 (a)H p−1 (b) 2

2

2

2

(mod p).

By combining the above, we get H(¯ a, ¯b) ≡ (21−w − 1)H(a, b) −

(−1)b H p−1 (a)H p−1 (b) 2 2 2w−1

(mod p).

Observe that H(a) + H(¯ a) ≡

2 H p−1 (a) 2 2w

(mod p).

The lemma follows from Thms. 8.2.3 and 8.2.7 immediately. Theorem 8.6.4. Let a, b ∈ N and a prime p ≥ a + b + 2. If the weight w = a + b is odd then we have   ? a w ζA (a, b) = ζ (a, b) = (−1) βw , (8.49) A1 1 a   w βw , (8.50) ζA1 (a, b) = (−1)a (2p−w − 1) a    ? 1−w a w ζA (a, b) = (2 − 1) 2 + (−1) βw , (8.51) 1 a ζA1 (a, b) = ζA1 (a, b) = (1 − 21−w )βw ,

? ζA (a, b) 1

? = ζA (a, b) 1

1−w

= (2

− 1)βw .

(8.52) (8.53)

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If a + b is even, then we have

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? ζA (a, b) = ζA1 (a, b) =0, 1 ? ζA (a, b) = ζA1 (a, b) = 1

(8.54) a

b

(2 − 2)(2 − 2) βa βb . 2w−1

(8.55)

Moreover,  ζA2 (2a, 2b) =

   (a − b)(1 − 2−2w ) 2w + 2 − w pβ2w+1 . 2w + 2 2a + 1

(8.56)

Proof. Equations (8.49) and (8.54) follow from Thm. 8.5.12. Equations (8.50) and (8.55) follow from Lemma 8.6.3. To prove (8.52), by stuffle relation we see that for (α, β) = (¯ a, b) or (a, ¯b), and odd w, we have 0 = ζA1 (α)ζA1 (β) = ζA1 (α, β) + ζA1 (β, α) + ζA1 (α ⊕ β)

= ζA1 (α, β) + sgn(αβ)(−1)w ζA1 (α, β) + ζA1 (α ⊕ β) = 2ζA1 (α, β) + ζA1 (α ⊕ β).

Thus 1 ζA1 (a, b) = ζA1 (a, ¯b) = − ζA1 (w) = (1 − 21−w )βw 2 ? from Thm. 8.2.7. Finally, all ζA identities are easy consequence of those 1 ? of ζA1 by the relation ζA1 (α, β) = ζA1 (α, β) + ζA1 (α ⊕ β). We now turn to Eq. (8.56). By superbity two reversal relation of Thm. 8.2.4

ζA2 (2a, 2b) = ζA2 (2b, 2a) + 2bpζA1 (2b + 1, 2a) + 2apζA1 (2b, 2a + 1). So by Eq. (8.51)   2(a − b)(1 − 2−2w ) 2w + 2 ζA2 (2a, 2b) = ζA2 (2b, 2a) + pβ2w+1 . 2a + 1 2w + 2

(8.57)

On the other hand, we have ζA2 (2a, 2b) + ζA2 (2b, 2a) = ζA2 (2a)ζA2 (2b) − ζA2 (2w) and therefore by Eqs. (8.15) and (8.29), ζA2 (2a, 2b) + ζA2 (2b, 2a) = −ζA2 (2w) = −2pwβ2w+1 . Equation (8.56) now follows easily from Eqs. (8.57) and (8.58).

(8.58)

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Turning to FES of depth 3, we first observe that for any α, β, γ ∈ Z

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H(α, β, γ) =H(α)H(β)H(γ) − H(γ)H(β, α) − H(γ)H(β ⊕ α) − H(γ, β)H(α) − H(γ ⊕ β)H(α) + H(γ, β, α)

+ H(γ ⊕ β, α) + H(γ, β ⊕ α) + H(γ ⊕ β ⊕ α).

This can be easily checked by the stuffle relations but the idea is hidden in the general framework set up by Hoffman [293]. Combining with the reversal relations one can obtain the following results without much difficulty. We leave its proof to the interested reader. Theorem 8.6.5. Let a, b, c ∈ N. If the weight w = a + b + c is even then 2ζA1 (a, b, c) = ζA1 (b + c, a) + ζA1 (c, a + b),

(8.59)

2ζA1 (a, b, c) = − ζA1 (a)ζA1 (c, b) + ζA1 (c, a + b) + ζA1 (b + c, a),

(8.60)

2ζA1 (a, b, c) = − ζA1 (c)ζA1 (b, a) − ζA1 (c, b)ζA1 (a)

+ ζA1 (b + c, a) + ζA1 (c, a + b).

(8.61)

If the weight w is odd then 2ζA1 (a, b, c) = ζA1 (c, a + b) + ζA1 (b + c, a) − ζA1 (a)ζA1 (c, b),

2ζA1 (a, b, c) = − ζA1 (c)ζA1 (b, a) − ζA1 (c, b)ζA1 (a)

+ ζA1 (a + b + c) + ζA1 (c, a + b).

(8.62) (8.63)

Because of the reversal relations when the weight is even there remains essentially only one more case to consider in depth 3. This is given by the next result. Theorem 8.6.6. Let a, b, c be positive integers such that w := a + b + c is even. Then for any prime p ≥ w + 3 we have   p−w+1 X  p−a k + c − 1 (1 − 2k )Bk Bp−w−k+1 ¯ H(c, b, a ¯) ≡ − p−w−k+1 c ak k=2    p−c X p−a k + c − 1 (1 − 2k )Bk B2p−w−k − 2p − w − k c ak k=p+1−b−c

−

(1 − 2p−c )(1 − 2p−a−b )Bp−a−b Bp−c . (a + b)c

Proof. The proof is basically a repeated application of Thm. 8.6.1. But we spell out all the details below because there are some subtle details that we need to attend to.

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By Eq. (8.2) and Fermat’s Little Theorem we have, modulo p,

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H(c, ¯b, a ¯) ≡ ≡

p−1 j−1 i−1 X X (−1)i X j p−1−b (−1) j k p−1−a c i i=1 j=1 k=1

p−1 X i=1

(−1) ic

i p−1−a X  k=0

 i−1 p − a Bk X (−1)j j ρ(k)+p−a−b−k , k p − a j=1

where ρ(k) = 0 if k < p − a − b and ρ(k) = p − 1 if k ≥ p − a − b (to make sure all exponents are positive in the sum of the second line above). By Lemma 8.2.5 p−1 p−1−a n  X X p − a Bk X n ¯ (−1)i in−r−c Fn,i,r H(c, b, a ¯) ≡ p − a r=0 r i=1 k k=0 p−1 p−1−a X n (1 − 2r+1 )Bk Br+1 X X p − a n−1 in−c−r ≡− r k (p − a)(r + 1) r=1 i=1 k=0   p−1 p−1−a X p − a (1 − 2n+1 )Bk Bn+1 X ((−1)i − 1)i−c , + (p − a)(n + 1) i=1 k k=0

where n = ρ(k) + p − a − b − k. Here we have used the fact that when Pp−1 r = 0 we have Fn,i,r = (−1)i−1 and thus the inner sum is i=1 in−c ≡ 0 (mod p) except when p − 1|n − c, i.e., except when n = c and k = p − w. But then Bk = 0 since w is even by assumption. Thus X p − a n  (1 − 2n−c+1 )Bk Bn−c+1 H(c, ¯b, a ¯) ≡ (p − a)(n − c + 1) k n−c 0≤k 1 is odd then both sums in Eq. (8.71) are Q-linear combinations of βλ1 s · · · βλr s for odd partitions Λ = (λ1 , . . . , λr ). If s = 1 then Eq. (8.71) is a Q-linear combination of q`2 βλ`+1 · · · βλr for odd partitions λ = ({1}` , λ`+1 , . . . , λr ) (λj > 1 for all j > `), where q`2 = ((2p−1 −1)/p)p∈P is an A1 -Fermat quotient. Proof. Apply Thm. 7.4.1 and then use Thm. 8.2.7. All are straightforward except in the case s = 1 we need the famous Claussen–von Staudt Theorem which says pβ1 = −1 in A1 (see [314, p. 233]). For example, by [538, Thm. 2.1] and [527, Thm. 5.2 ] we have 2 3 ζA3 (¯ 1) = −ζA3 (1) + H p−1 (1) = −2q2 + pq22 − p2 q32 − p2 β3 . 2 3 4

(8.72)

Observe that Odd(2) = {(1, 1)}, Odd(3) = {(1, 1, 1), (3)}, Odd(4) = {(1, 1, 1, 1), (1, 3)}. It is obvious that c(1,··· ,1) = 1, c(3) = 2, c(1,3) = 8, c(1,1,3) = 20, and c(1,1,1,3) = c(3,3) = 40, which implies that
2
1 4 ζA1 {¯ 1}3 = − q32 − β3 , ζA1 {¯1}4 = q42 + q2 β3 , 3 2 3

4 4 2 1 5 2 6 3 5 6 ζA1 {¯ 1} = − q2 − q2 β3 , ζA1 {¯1} = q + q β3 + β32 . 15 45 2 3 2 8 The expression in the examples above are unique if we assume Conjecture 8.1.5.
ζA1 {¯ 1}2 = 2q22 ,

8.7 Historical Notes The following identity is one of the series related to π that Ramanujan listed in [480]:  3 ∞ X −1/2 2 (4k + 1) = . k π k=0

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This was actually discovered earlier by Bauer in 1859. In his seminal paper [557] of 1997 van Hamme studied the (p − 1)-st partial sums of over a dozen such series for odd primes p. Among the many conjectures he proposed one can find the following super congruence: (p−1)/2

X k=0

(4k + 1)

  3  −1/2 −1 p ≡ p k

(mod p3 ).

This was later settled by Mortenson [428] using the Gamma function and the hypergeometric functions. Since then this kind of congruences have been said of van Hamme type. The Wolstenholme’s Theorem was improved by Glaisher [239] to the super congruence p−1 X 1 1 ≡ − p2 Bp−3 k 3

(mod p3 ) for all prime p > 3,

(8.73)

k=1

and by Z.-W. Sun [527] as follows:   p−1 X 2Bp−3 1 B2p−4 2 ≡p − k p−3 2p − 4

(mod p4 ) for all prime p > 5.

(8.74)

k=1

Ever since Wolstenholme published his work on the congruences Eq. (8.1) a plethora of generalizations are given along various approaches, the majority of which involve variations with composite moduli or changes of the summation range, see, for example, the work by Leudesdorf [388], Glaisher [238, 239], Vandiver [558], Bayat [43], Pan [465] and so on. For a more complete history, please refer to the survey article [426] by Mestrovic. Virtually none of these work, however, treated multiple variable cases. In an unpublished work R. Bruck seems to be the first who studied the congruence properties of the homogeneous multiple harmonic sums. Generalizing Wolstenholme’s Theorem to the more general MHSs (i.e., the partial sums of the MZVs) was the initial motivation of [609,619] by the author starting around 2002. For example, Thm. 8.5.12, 8.5.13, and 8.5.17 were proved in [619] as Thm. 3.1, 3.5, and 3.2, respectively. A little later, Hoffman independently studied these sums by his powerful algebraic setup using word algebras. In order to find a suitable theoretical framework in which this kind of results can be organized and further investigated, the author first defined in [625] an ad`ele-like structure essentially the same as Eq. (8.3) which was later defined, when the superbity is 1, by Kaneko and Zagier who called it poorman’s ad`ele.

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The first systematic investigation of congruences involving the AMHSs was carried out in the joint work [539]. We have recasted these sums as the finite Euler sums in this chapter due to their close relations to the classical Euler sums. Further generalization of these to some finite version of multiple polylogs can be found in [495]. Note that [578] contains one more statement which is equivalent to the two congruences in Wolstenholme’s Thm. 8.2.1:     2p − 1 2p 3 ≡ 1 (mod p ), i.e., ≡ 2 (mod p3 ). p−1 p Wolstenholme’s Theorem was revisited by Gardiner [234] who also showed that the three statements are equivalent. In [508] Sekine extended some of the author’s results of [609, 619] to the partial sums truncated at p2 instead of p. A further study of congruence properties of MHS with varying summation ranges was carried out in [627] based on the earlier works of Boyd [78] and Eswarathasan and Levine [209]. Theorem 8.2.7 can also be obtained from [538, Thm. 2.1] combined with [527, Thm. 5.2]. Note that both terms in [538, Thm. 2.1] contribute nontrivially when s is even since the modulus is p2 . In this case, the theorem is also given by [123, (6.2)]. For the FMZVs, as an improvement of Thm. 8.5.1 Zhou and Cai [637] obtained the super congruences in Thm. 8.5.4. Neither FMZV dualities given by Thm. 8.5.10 (ϕ-dual) and Thm. 8.5.11 (v-dual) is the same as that of the MZVs. These were discovered by Hoffman first in [292, 294]. Definition 8.5.6 comes from Hoffman’s modification of an idea of Vermaseren described in [560, Appendix B]. Our proof of Lemma 8.5.8 here follows that of [292, Thm. 4.2] while Thm. 8.5.11 is [619, Thm. 2.11]. J. Rosen [486] extended this result to an asymptotic type formula with weights associated with the powers of p. This hints at a connection to the p-adic MZVs subsequently studied by Jarossay [317–319]. In particular, the shuffle relation of the SMZVs stated in Thm. 6.3.4 corresponds to the shuffle relation satisfied by the FMZVs. These are given precisely by Thm. 8.4.3, which also includes the case for FESs. In the spring of 2013 after the author gave an Ober-Seminar talk at the Max Planck Institute for Mathematics at Bonn, Prof. D. Zagier mentioned that he had come to the dimension part of Conjecture 8.5.14 some time earlier [602]. He also informed the author of Conjecture 8.5.15 which was to appear in a joint paper with Kaneko [336]. Note that Thm. 6.4.19, proved by Yasuda [585], shows that the map in Conjecture 8.5.15 is surjective. In order to find the upper bounds of the FMZV and FES dimensions, it is

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essential to determine the correct analog of the DBSFs. This was carried out in [633] which is incorporated into Sections 8.4-8.6. A statement of the shuffle relations of the FMZVs similar to that of Thm. 8.4.3 first appeared in the Japanese survey paper [330] by Kaneko. Theorem 8.5.23, an FMZV analog of the sum formula for the SMZVs (see Thm. 6.2.2) was proved by Saito and Wakabayashi in [492, 493], who also found the results in Exercise 8.10 which are the sum formulas for FMZVs in the same spirit as that of Conjecture 5.10.2 (for MZVs) or that of Thm. 10.9.3 (for MZ? Vs). The FMZV analog of the Ohno’s relation of the MZVs (see Thm. 5.5.9) can be found in Exercise 8.11, which was conjectured by Kaneko and proved by Oyama [464]. In [312] Imatomi et al. expressed the FMZVs as the multi-poly-Bernoulli numbers (see Eq. (5.117) for the definition):   ζA1 (s1 , s2 , . . . , sd ) = − Cp−2 (s1 − 1, s2 , . . . , sd ) (mod p) . p∈P

Theorems 8.6.11 and 8.6.10 were proved by Hessami Pilehroods and Tauraso in [278], which led to the improvement of the upper bounds for the dimensions of FMZ7,1 and FMZ9,1 (see [278, Cor. 4.4]). This yields that dim FMZ7,1 ≤ 1 and dim FMZ9,1 ≤ 2 (see Table 8.2). Exercises 8.1. For every positive integer `, prove that the following procedure provides an embedding ι` : R ,→ A` as sets. If r ∈ Q then ι` (r) is defined as in Eq. (8.4). Assume r = n.d1 d2 · · · is irrational where n is its integral part and d1 , d2 , . . . are its decimal digits. Let r
k = n.d1 d2 . . . dk be the kth approximation of r. We define ι` (r) = ιpj ,` (r) j≥1 by the following. ιpj ,` (r) = ιpj ,` (rk ) for all j = k(k − 1) + 1, k(k − 1) + 2, · · · , k(k + 1). Roughly speaking, this means that for each k we put the kth approximation rk of r at 2k consecutive components. 8.2. Prove Lemma 8.5.2 using Eq. (8.2) and Fermat’s little Theorem: kp

2

(p−1)

≡1

(mod p3 )

for all k not divisible by p. You also need the fact that pB2m is p-integral for all m ∈ N by von Staudt–Clausen Theorem [314, p. 233]. 8.3. Prove the case a = 0 of Eq. (8.69) using Eq. (7.24), Thm. 8.5.1, and Thm. 8.6.4. 8.4. Prove Eq. (8.46) using the same idea as in the proof of Eq. (8.45).

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8.5. Prove Eq. (8.46) by backward induction on d by proving the following key identity first: Sw,d,1 =

     d−1 X w−d+j−1 ? w−1 (−1)j Sw,d−j,1 = 1 + (−1)d βw . j d−1 j=0

8.6. Compute tog(z¯4 z2 z1 z¯3 z2 z¯2 z¯1 z7 z¯6 ) and tog−1 (b1 a2 b1 a3 b0 b1 a4 b0 a4 b1 ab0 ). 8.7. Prove Prop. 8.6.8 by mimicking the proof of Prop. 6.1.3 and Thm. 6.3.2. You will need regularized Euler sums in two ways from Chap. 13 given by Prop. 13.3.8 and apply their relation given by Thm. 13.3.31. 8.8. Let a, b ∈ N0 , a ˜ = 2a + 1 and ˜b = 2b + 3. Then show that ? ζA ({2}a , 3, {2}b , 3) =2ζA1 (˜ a, 1, ˜b + 1) − 6ζA1 (˜b + 1, a ˜, 1) 1 ˜ + (8a − 6)ζA (˜ a + 1, b, 1) + 16ζA (˜ a + ˜b, 1, 1) 1

1

+ (8b + 14)ζA1 (˜ a, ˜b + 1, 1) − 16ζA1 (1, a ˜, ˜b, 1) − 2ζA (˜ a + 1, 1, ˜b) − 2ζA (˜b, a ˜ + 1, 1) + 4ζA (˜ a, ˜b, 2). 1

1

1

8.9. Verify that dim ES2 ≤ 2 and dim ES3 ≤ 3 by using the linear DBSFs and reversal relations for FESs. Are DBSFs enough? 8.10. Suppose a and b are odd positive integers and c is an even positive integer. Show that for all nonnegative integers m and n with (m, n) 6= (0, 0), we have X ζA1 ({c}n0 , a, {c}n1 , b, {c}n2 , . . . , a, {c}n2m−1 , b, {c}n2m ) = 0, P2m i=0 ni =n n0 ,...,n2m ≥0

X P2m

i=0 ni =n n0 ,...,n2m ≥0

? ζA ({c}n0 , a, {c}n1 , b, {c}n2 , . . . , a, {c}n2m−1 , b, {c}n2m ) = 0. 1

8.11. Let s = (s1 , . . . , sd ) ∈ Nd and suppose its v-dual s∨ has depth d0 . Then for any n ∈ N0 , we have X X
ζA(s + e) = ζA (s∨ + e0 )∨ . e∈(N0 )d ,|e|=n

e0 ∈(N0 )d0 ,|e0 |=n

You may want to read Sec. D.3 first before trying the following problems.

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8.12. With your favorite computer algebra system, verify the following relations which give some evidence for Conjecture 8.6.9: ζA1 ({¯ 1}3 ) = 4ζA1 (¯ 2, 1) + 4ζA1 (¯1, 1, 1), 4 ζA1 ({¯ 1} ) = 4ζA1 (¯ 1, 1, 2) + 4ζA1 (¯1, 2, 1) + 8ζA1 (¯1, {1}3 ), and ¯ 1) + 4ζ∗S(1, ¯ 1, 1) (mod ζ(2)), ζ∗S({¯ 1}3 ) ≡ 4ζ∗S(2, S ¯ 4 S ¯ ζ∗ ({1} ) ≡ 4ζ∗ (1, 1, 2) + 4ζ∗S(¯1, 2, 1) + 8ζ∗S(¯1, {1}3 )

(mod ζ(2)).

Note that the verification can be done at two levels, numerically using congruences, or rigorously using the linear DBSFs and reversal relations for FESs (DBSFs are not enough!). 8.13. With your favorite computer algebra system, verify numerically that ζA1 (s) = 0 if  q
(1) s = {1}m , 2, {1}n , 2 , {1}m , 2, {1}n for q, m, n ≥ 0, where either (i) q is odd, or(ii) q is even and m + n is even. n
(2) s = {2}m , 3, {2}m for m, n ≥ 0. Can you prove these rigorously? 8.14. With your favorite computer algebra system, verify numerically the following identities in A2 : (setting ζ2 = ζA2 ) 68ζ2 (2, 3, 1, 1) = − 480ζ2 ({1}5 , 2) + 716ζ2 ({1}3 , 4) − 843ζ2 (1, 6)

34ζ2 (3, 1, 1, 2) = − 585ζ2 ({1}5 , 2) + 998ζ2 ({1}3 , 4) − 1029ζ2 (1, 6) 68ζ2 (3, 3, 1) = 1730ζ2 ({1}5 , 2) − 2480ζ2 ({1}3 , 4) + 2319ζ2 (1, 6)

12ζ2 (5, 1, 1, 1) = − 73ζ2 ({1}6 , 2) + 12ζ2 (1, 1, 6) − 3ζ2 (1, 2, 5)

84ζ2 (3, 2, 1, 3) = − 995ζ2 (1, 8) + 952ζ2 ({1}3 , 6) − 1288ζ2 ({1}5 , 4) − 437ζ2 ({1}7 , 2) − 624ζ2 (1, 2, 6)

924ζ2 (2, 1, 3, 3) = 2509ζ2 (1, 8) − 6356ζ2 ({1}3 , 6) + 5432ζ2 ({1}5 , 4) − 180ζ2 ({1}7 , 2) + 801ζ2 (1, 2, 6)

924ζ2 (2, 3, 1, 3) = − 9424ζ2 (1, 8) − 5824ζ2 ({1}3 , 6) + 11368ζ2 ({1}5 , 4) − 3807ζ2 ({1}7 , 2) + 852ζ2 (1, 2, 6)

36ζ2 (1, 2, 5, 1) = 358ζ2 (1, 8) − 248ζ2 ({1}3 , 6) + 464ζ2 ({1}5 , 4) + 5ζ2 ({1}7 , 2) + 147ζ2 (1, 2, 6).

Can you prove any of these rigorously?

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8.15. With your favorite computer algebra system, verify numerically the following identities in A2 : (setting ζ2 = ζA2 ) ¯ 1), 4ζ2 (1, ¯ 1) = − 3ζ2 (2) − 4ζ2 (1, ¯ 1) ¯ = 9ζ2 (2) + 16ζ2 (¯1, 1), 8ζ2 (1, 4ζ2 (¯2) = 3ζ2 (2), 2ζ2 (1, 1) = − ζ2 (2). Can you prove all of these rigorously? These would imply that dim ES2,2 ≤ 2. Similarly, verify numerically ζ2 (1, 2) = 2ζ2 (¯ 3), ¯ 1) = ζ2 (3) ¯ − 2ζ2 (¯1, 2) − 4ζ2 (¯1, 1, 1), 2ζ2 (1, 1,

4ζ2 (¯ 1, ¯ 1, 1) = 3ζ2 (¯ 3) + 6ζ2 (¯1, 2) + 12ζ2 (¯1, 1, 1), ¯ 2ζ2 (2, 1) = − ζ2 (¯ 3), ζ2 (2, 1) = − 2ζ2 (¯ 3), ¯ ¯ 2ζ2 (1, 2) = − ζ2 (3), ζ2 (¯ 1, −2) = ζ2 (¯ 1, 2) − ζ2 (¯3),

16ζ2 (1, ¯ 1, ¯ 1) = − 17ζ2 (¯ 3) − 26ζ2 (¯1, 2) − 2ζ2 (¯1, ¯1, ¯1) − 40ζ2 (¯1, 1, 1), 4ζ2 (¯ 2, ¯ 1) = ζ2 (¯ 3) + 14ζ2 (¯1, 2) − 6ζ2 (¯1, ¯1, ¯1) + 24ζ2 (¯1, 1, 1), ζ2 (1, 1, ¯ 1) = ζ2 (¯ 3) − 2ζ2 (¯1, 2) + ζ2 (¯1, ¯1, ¯1) − 3ζ2 (¯1, 1, 1), 16ζ2 (¯ 1, 1, ¯ 1) = 17ζ2 (¯ 3) − 70ζ2 (¯1, 2) + 26ζ2 (¯1, ¯1, ¯1) − 104ζ2 (¯1, 1, 1), ζ2 (2, ¯ 1) = 5ζ2 (¯ 1, 2) − 2ζ2 (¯3) − 2ζ2 (¯1, ¯1, ¯1) + 8ζ2 (¯1, 1, 1).

These would imply that dim ES3,2 ≤ 4.

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Chapter 9

q -Analogs of Multiple Harmonic (Star) Sums

We have seen from the preceding two chapters that the MHSs and MH? Ss not only have a lot of nice properties but also are closely intertwined with the MZVs and the MZ? Vs. In this chapter, we move on to study their qanalogs which again will have important applications in a number of results concerning the MZVs and the MZ? Vs. 9.1 Definitions and Notation Throughout this chapter let m and n denote nonnegative integers and q a real number with 0 < q < 1. For any real number a, we have the quantum factorial and quantum Pochhammer symbol defined by: (a)0 := (a; q)0 := 1,

(a)n := (a; q)n :=

n−1 Y k=0

(1 − aq k ),

n ≥ 1.

As a convention throughout the chapter, we always use [ ] to represent q-analog objects. For example, the q-analog of a positive integer n is defined as [n] = [n]q :=

n−1 X

qk =

k=0

and the Gaussian q-binomial coefficient  (q)n      , n n = := (q)m (q)n−m  m m q 0,

1 − qn , 1−q

if 0 ≤ m ≤ n, otherwise.

Recall that the double cover D0 of N0 is defined in Definition 7.1.1. For m ∈ N0 , t = (t1 , . . . , tm ) ∈ Zm and s = (s1 , . . . , sm ) ∈ Dm 0 , we set s = ∅ if 279

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m = 0 and define the q-analogs of multiple harmonic (star) sums (q-MHS or q-MH? S) by X

Hnt [s] :=

n≥k1 >···>km

X

Hn?,t [s] :=

n≥k1 ≥···≥km

m Y sgn(sj )kj q kj tj , [kj ]|sj | ≥1 j=1 m Y sgn(sj )kj q kj tj , [kj ]|sj | ≥1 j=1

with the convention that Hnt [s] = 0 if n < m, and Hn?,t [∅] = Hnt [∅] = 1 for all n ≥ 0. Note that we allow sj to be 0 or ¯0 in these sums. For the starversion, since we will consider only Hn?,1 [s] in this chapter, we will simply write Hn? [s] = Hn?,1 [s], namely, X

Hn? [s] =

n≥k1 ≥···≥km

m Y sgn(sj )kj q kj . [kj ]|sj | ≥1 j=1

Now we fix a symbol θ and for any r ∈ Z ∪ {θ} and k ∈ N, define  if r > 0;  rk(k − 1)/2, Q(r, k) := rk(k − 1)/2 − k, if r ≤ 0;  0, if r = θ. Let m ∈ N0 . For any s = (s1 , . . . , sm ) ∈ Dm 0 and two mollifiers t = m (t1 , . . . , tm ) ∈ (N0 ) and r = (r1 , . . . , rm ) ∈ (Z ∪ {θ})m , we define the mollified companion of Hn1 [s] by n m Y q tj kj +Q(rj ,kj ) (1 + q kj ) X k1 Hn [s; t; r] := , (9.1) n+k  1 sgn(sj )kj [kj ]|sj | k j=1 n≥k >···>k ≥1 1

m

1

where we set Hn [s; t; r] = 1 if m = 0. We call (s; t; r) an admissible triple of mollifiers if the limit limn→∞ Hn [s; t; r] exists. Definition 9.1.1. Let
be a non-commutative binary operation on Z ∪ {θ}, called -plus, such that • θ
a = a
θ = a for all a ∈ Z ∪ {θ}, • a
b = a + b for all a, b ∈ Z with (a, b) 6= (1, −1), (−1, 1), and • 1
(−1) = θ and (−1)
1 = 0. Lemma 9.1.2. Let r ∈ {θ} ∪ Z \ {0}, d ∈ {0, −1}. Then for any k ∈ Z Q(r, k) + Q(1, k) = Q(r
1, k), 2

k + Q(d, k) = Q(2
d, k).

(9.2) (9.3)

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Moreover, the projection

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π : (Z ∪ {θ},
) −→ (Z, +) a 7−→ a

θ 7−→ 0,

∀a ∈ Z,

is a homomorphism of semi-groups and its restriction to Z∗ is injective. Proof. These follow from straightforward computation. For an admissible triple of mollifiers (s; t; r)] of depth m, we define n o s1 ◦ · · · ◦ sm ; t1 ◦ · · · ◦ tm ; r1 ◦ · · · ◦ rm n o ≈ ≈ e ; p) ∈ Dm × Zm × (Z ∪ {θ})m : (p; p e ; p)  (s; t; r) := (p; p to be the set of triples of strings produced by replacing every ◦ in s by either comma “,” or O-plus “⊕” (see Eq. (7.1)), replacing every ◦ in t by either comma “,” or the usual plus “+”, and replacing every ◦ in r by either comma “,” or -plus “
”. Moreover, the commas should be at the same positions for all the three compositions s, t and r. Finally, to save space, we define m  a◦k = |a ◦ .{z . . ◦ a}, and Cat aj } = (a1 , . . . , am ). j=1 k times

9.2 Some q-Combinatorial Identities We start with a preliminary combinatorial lemma. Lemma 9.2.1. For integers n ≥ 1 and l ≥ 0, we have n n n X l [l] − [n] l k k (k k ) 2 (1 + q ) n+k (−1) q = n+l (−1)l q (2) , [n] k l k=l+1 n n n X 2 k(k−1) k l (1 + q k )[k] n+k q = ([n] − [l])  n+l  ql , k

k=l+1

n X 1 + qk k=1

[k]

Moreover, if l ≥ 1 then

(9.4) (9.5)

l

n

2

k n+k  qk = k

n X qm . [m] m=1

k n n X qk ql l l   = 2 n+l . [k]2 k+l [l] l l k=l

(9.6)

(9.7)

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Proof. The general idea of the proof is to use WZ-method. First, for all positive integers k, l and n, we define n k q n+k − 1 k k G1 (n, k) := k F1 (n, k), F1 (n, k) :=(1 + q ) n+k  (−1)k−1 q (2) , q +1 k n 1 − q n+k k F2 (n, k), F2 (n, k) := n+k G2 (n, k) :=  (1 − q 2k )q k(k−1) , 1 − q 2k k l 1 + q k k2 q l−k+1 1 − q k k F3 (l, k) := l+k q , G (l, k) := F3 (l, k),  3 1 − qk 1 + q k 1 − q l−k+1 k l q k−l k (1 − q k−l )(1 − q k+l ) F4 (l, k). F4 (l, k) := 2 l+k , G4 (l, k) := [k] q k−l k Set the difference operator ∆n f (n) := f (n + 1) − f (n). Easy computation shows that (1 − q n )F1 (n, k) =∆k G1 (n, k),

F2 (n, k) = − ∆k G2 (n, k),

l 2

(1 − q ) F4 (l, k) =∆k G4 (l, k),

∆l F3 (l, k) = − ∆k G3 (l, k).

Therefore (1 − q n )

n X

n X

F1 (n, k) =

k=l+1

k=l+1

∆k G1 (n, k) = −G1 (n, l + 1)

n 1 − q n+l+1 n−l l = F1 (n, l + 1) = (1 − q ) n+l  (−1)l q l(l+1)/2 , 1 + q l+1 l n n X l F2 (n, k) = G2 (n, l + 1) = n+l  (1 − q n )q l(l+1) , l

k=l+1 n−1 X l=0

∆k G3 (l, k) = − l 2

(1 − q )

n X

n−1 X l=0

F4 (l, k) =

∆l F3 (l, k) = F3 (0, k) − F3 (n, k) = −F3 (n, k),

n X k=l

k=l

n l ∆k G4 (l, k) = G4 (l, n + 1) = (1 − q) n+l . 2

l

So Eqs. (9.4), (9.5) and (9.7) are proved. Moreover, n X k=1

F3 (n, k) = −

n−1 X

n X

∆k G3 (m, k) =

m=0 k=1

n−1 X

(G3 (m, 1) − G3 (m, n + 1))

m=0

=

n−1 X m=0

G3 (m, 1) =

n−1 X

q m+1 , 1 − q m+1 m=0

which implies Eq. (9.6). This completes the proof of the lemma.

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283

Lemma 9.2.2. Let a ∈ D0 , b ∈ N0 , c ∈ N, r ∈ {θ} ∪ Z \ {0}, and (x; y; z) an admissible triple of mollifiers. Then for any positive integer n, X 1 ≈ e , y; p, z]. Hn [a, x; b, y; r
1, z] = Hn [p, x; p c [n] ≈ (p;e p;p)

≈


e ; p)  0, {1}c−1 , a ⊕ 1; {0}c , b; 1, {θ}c−1 , r in the sum. where (p; p Proof. We prove the lemma by induction on c. Suppose all the mollifiers x, y, z have depth m. Set n k An,k = (−1)k (1 + q k )q k(k−1)/2 n+k .

(9.8)

k

Then by Eq. (9.4),   n (1 + q l ) X 1 1 An,k = − An,l [l] [n] [l]

(9.9)

k=l+1

which, together with Eq. (9.2) yields Hn [0, a ⊕ 1, x; 0, b, y; 1, r, z] =

X n≥k>k0 >k1 >···>km

=

X n≥k0 >···>km

m An,k q bk0 +Q(r,k0 ) (1 + q k0 ) Y q yj kj +Q(zj ,kj ) (1 + q kj ) (− sgn(a))k0 [k0 ]|a|+1 j=1 sgn(xj )kj [kj ]|xj | ≥1

m n q bk0 +Q(r,k0 ) (1 + q k0 ) Y q yj kj +Q(zj ,kj ) (1 + q kj ) X An,k (− sgn(a))k0 [k0 ]|a|+1 j=1 sgn(xj )kj [kj ]|xj | k=k +1 ≥1 0

1 = Hn [a, x; b, y; r
1, z] − Hn [a ⊕ 1, x; b, y; r
1, z]. [n] This proves the lemma for c = 1. Now suppose c > 1. By the case c = 1 we have just proved,   1 1 1 H [a, x; b, y; r
1, z] = H [a, x; b, y; r
1, z] n n [n]c [n]c−1 [n] 1 1 Hn [a ⊕ 1, x; b, y; r
1, z] + c−1 Hn [0, a ⊕ 1, x; 0, b, y; 1, r, z]. = c−1 [n] [n] For the first summand, we now apply induction assumption using case c − 1 with a replaced by a ⊕ 1. For the second summand, we apply case c − 1 of Lemma 9.2.2 with a = 0, b = 0, and r = θ. Hence the above is equal to X ≈ e , y; p, z] Hn [p, x; p
≈ (p;e p;p) 0,{1}c−2 ,(a⊕1)⊕1;{0}c−1 ,b;1,{θ}c−2 ,r

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X

+ ≈

(p;e p;p) 0,{1}c−1 ;{0}c ;1,{θ}c−1




e , b, y; p, r, z] Hn [p, a ⊕ 1, x; p ≈

X

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e , y; p, z], Hn [p, x; p

≈

(p;e p;p) 0,{1}c−1 ,a⊕1;{0}c ,b;1,{θ}c−1 ,r




since (a ⊕ 1) ⊕ 1 = 1 ⊕ (a ⊕ 1) for all a ∈ D0 . We have now completed the proof of the lemma. The next corollary is the degenerate case of the preceding lemma. Corollary 9.2.3. Let c ∈ N0 and Π(c) = (0, {1}c ; {0}c+1 ; 1, {θ}c ). Then X 1 ≈ e ; p]. =− Hn [p; p c [n] ≈ (p;e p;p)Π(c)

Proof. The case c = 0 follows from by Eq. (9.4) by setting l = 0. For c ≥ 1, using the c = 0 case we get X 1 1 ≈ e ; p] = − c Hn [¯0; 0; 1] = − Hn [p; p c [n] [n] ≈ (p;e p;p)Π(c)

by taking a = ¯ 0, b = 0, r = θ and x = y = z = ∅ in Lemma 9.2.2. Hence the corollary is proved. 9.3 q -MH? S Identities In this section we prove some general rules which explain what to expect for the q-MH? S Hn? [s] when we add an argument component in front of the positive composition s. This allows us to extend expansion formulas from the two base cases of depth one, namely Eqs. (9.10) and (9.11), to an arbitrary positive composition s. ≈ e ; p) to be For simplicity, throughout this section we always set p = (p; p a triple of mollifiers. Theorem 9.3.1. Let n, c ∈ N, c ≥ 2 and Π1 (c) = (2, {1}c−2 ; 1, {0}c−2 ; 1, {θ}c−2 ). Then n n X 2 1 + qk k ? Hn [1] = Hn [1; 1; 2] = (9.10) n+k q k , [k] k k=1 X Hn [p]. Hn? [c] = − (9.11) pΠ1 (c)

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Proof. Equation (9.10) is just Eq. (9.6). To prove Eq. (9.11) we proceed by induction on n. When n = 1 we have H1? [c] = q. On the other hand, X H1 [p] = H1 [c; 1; 1] = −q, pΠ1 (c)

since the depth of every mollifier in p must be 1 in order to have nontrivial contribution. Therefore the formula is true when n = 1. Suppose the statement is true for n − 1. Then by the definition qn ? Hn? [c] = Hn−1 [c] + . [n] Applying inductive hypothesis, we obtain X qn (9.12) Hn? [c] = − Hn−1 [p] + c . [n] pΠ1 (c)

Set Π0 (c) = (0, {1}c−2 ; {0}c−1 ; {θ}c−1 ). To save space, for any string λ = (λ1 , . . . , λm ) we write the substring λb1 = (λ2 , . . . , λm ). Then the sum on the right-hand side of Eq. (9.12) becomes X X ≈ ≈ eb1 ; 1
p 1 , pb1 ] Hn−1 [p] = Hn−1 [2 + p1 , pb1 ; 1 + pe1 , p pΠ0 (c)

pΠ1 (c)

=

X

X

pΠ0 (c) n>k1 >...>km ≥1

m q k1 (1+ep1 ) An−1,k1 Y q pej kj (1 + q kj ) , [k1 ]2+p1 [kj ]pj j=2

where An,k is defined by Eq. (9.8). Plugging this into Eq. (9.12) and using the formula   [k]2 An−1,k = An,k 1 − 2 q n−k , (9.13) [n] we obtain m X X q k1 (ep1 +1) An,k1 Y q pej kj (1 + q kj ) Hn? [c] = − [k1 ]2+p1 [kj ]pj 0 j=2 pΠ (c) n≥k1 >···>km ≥1

+

qn [n]2

X

X

pΠ0 (c) n≥k1 >···>km ≥1

m qn q k1 pe1 An,k1 Y q pej kj (1 + q kj ) + , [k1 ]p1 [kj ]pj [n]c j=2

which implies Hn? [c] = −

X pΠ1 (c)

Hn [p] +

qn [n]2

X pΠ00 (c)

Hn [p] +

qn , [n]c

where Π00 (c) = (0, {1}c−2 ; {0}c−1 ; 1, {θ}c−2 ). Hence the theorem follows from Cor. 9.2.3 immediately since Π1 (c) = Π(c − 2) there.

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We now turn to the general rules. For any string λ = (λ1 , . . . , λm ), we set λb1 = (λ2 , . . . , λm ) as above. Theorem 9.3.2. Let n ∈ N and s = (s1 , . . . , sd ) ∈ Nd . Set δ(s) = 1 if sd = 1 and δ(s) = −1 if sd > 1. Suppose there is a triple of mollifiers ≈ ≈ ≈ e λ) = (λ1 , . . . , λm ; λ e1 , . . . , λ em ; λ 1 , . . . , λ m ) satisfying λ := (λ; λ; ≈

≈

λ 1
· · ·
λ j ∈ {1, 2}

∀j ≥ 1,

(9.14)

such that there is an expansion of the form X Hn? [s] = δ(s) Hn [p]. p λ

Then for any positive integer c ∈ N, we have X Hn [p], Hn? [c, s] = δ(s)

(9.15)

pΠc (λ) ≈

≈

e 2, λ 1
(−2), λb), Π2 (λ) = (2 ⊕ λ1 , λb; 1 + where1 Π1 (λ) = (1, λ; 1, λ; 1 1 ≈ ≈ e1 , λ e b; λ), and Πc (λ) = (2, {1}c−3 , λ1 ⊕ ¯1, λb; 1, {0}c−3 , λ; e 1, {θ}c−3 , λ 1
λ 1 1 ≈

(−1), λb1 ) for c ≥ 3. Moreover, in all the sets Πc (λ) the third component still satisfy Eq. (9.14).

Note that the condition in Eq. (9.14) essentially guarantees that all the triples of mollifiers considered in this section are admissible. Proof. Write Πc = Πc (λ) throughout the proof. We prove the identity in the theorem by induction on n. When n = 1 the theorem is clear. Assume Eq. (9.15) is true for n − 1. First, we prove Eq. (9.15) for c = 1. By the definition, we have qn ? ? Hn? [1, s] = Hn−1 H [s]. [1, s] + [n] n Applying induction assumption, we obtain X qn X δ(s)Hn? [1, s] = Hn−1 [p] + Hn [p]. (9.16) [n] pΠ1

≈

e λ1
Setting Π0 = (λ; λ; (k1 , . . . , km ) we write

≈ (−2), λb1 ).

p λ

To save space, for p  Π0 and k = ≈

sgn(pj )kj q pej kj +Q(p j ,kj ) (1 + q kj ) fj (p, k) = . [kj ]|pj | 1 We have abused the notation here by writing (1, λ; . . . ) for the concatenation (1, λ1 , . . . , λm ; . . . ). No confusion should arise.

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pΠ1

=

X  pΠ0

 ≈ ≈ ≈ e ; 2, p] + Hn−1 [1 ⊕ p1 , pb1 ; 1 + pe1 , p eb1 ; 2
p 1 , pb1 ] Hn−1 [1, p; 1, p

 =

n−1

X

k0 n−1+k  0

X 

pΠ0

k0

n>k0 >k1 >···>km ≥1

2 m (1 + q k0 )q k0 Y fj (p, k) [k0 ] j=1

 ≈ m k1 (1+e p1 )k1 +Q(2
p 1 ,k1 ) Y (1 + q )q k1 fj (p, k) . n−1+k  k1 [k ]1+|p1 | 1 sgn(p ) 1 1 k j=2 ≥1 n−1

+

X n>k1 >···>km

1

Plugging the above expression into Eq. (9.16) and using Eq. (9.13), we obtain qn X Hn [p] δ(s)Hn? [1, s] − [n] p λ  n 2 m X X (1 + q k0 )q k0 Y k0  = fj (p, k) n+k0  [k0 ] k0 j=1 pΠ0 n≥k0 >k1 >···>km ≥1 n m X Y qn k0 k0 k0 (k0 −1) − 2 (1 + q )q [k ] fj (p, k) n+k  0 0 [n] k0 j=1 n≥k0 >k1 >···>km ≥1 n ≈ m X (1 + q k1 )q (1+ep1 )k1 +Q(2
p 1 ,k1 ) Y k1 fj (p, k) + n+k1  sgn(p1 )k1 [k1 ]2a+1+|p1 | k1 j=2 n≥k1 >···>km ≥1  n ≈ m n k1 p e1 k1 +Q(2
p 1 ,k1 ) Y X q (1 + q )q k1 − 2 fj (p, k) . n+k  1 [n] sgn(p1 )k1 [k1 ]|p1 |−1 k j=2 n≥k >···>k ≥1 1

m

1

Noting that the first and third sums on the right-hand side add up to X Hn [p], we have pΠ1

X qn X Hn [p] + Hn [p] [n] pΠ1 p λ  m n X X Y X  fj (p, k)

δ(s)Hn? [1, s] = qn − 2 [n]

pΠ0 n≥k1 >···>km ≥1

j=1

n

k0 n+k  0 k0 k0 =k1 +1

 ≈ m k1 p e1 k1 +Q(2
p 1 ,k1 ) Y (1 + q )q k1 + n+k fj (p, k) .  k1 [k ]|p1 |−1 1 sgn(p ) 1 1 k j=2 n 1

k0

(1 + q k0 )q 2( 2 ) [k0 ]−1

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Summing the multiple sum in the above over k0 by Eq. (9.5) and noticing ≈ that for p  Π0 , by Eq. (9.14), the first component p 1 can take only values equal to −1 and 0, with the help of Eq. (9.3) we obtain X Hn [p]. δ(s)Hn? [1, s] = pΠ1

This proves Eq. (9.15) in the case c = 1 by induction. When c = 2, by the definition we have ? Hn? [2, s] = Hn−1 [s] +

qn ? H [s]. [n]2 n

The induction assumption implies that X qn X δ(s)Hn? [2, s] = Hn [p]. Hn−1 [p] + 2 [n] pΠ2

(9.17)

p λ

Expanding the first sum on the right-hand side, we get X X ≈ eb1 ; p] Hn−1 [p] = Hn−1 [2 ⊕ p1 , pb1 ; 1 + pe1 , p pΠ2

p λ

n−1 =

k1 n−1+k  1 k1 p λ n>k1 >···>kr ≥1

X

X

r q k1 Y fj (p, k). [k1 ]2 j=1

Hence δ(s)Hn? [2, s] =

qn X Hn [p] [n]2 p λ

+

X

X

p λ n≥k1 >···>kr ≥1

n   r q k1 qn Y k1 − 2 fj (p, k) n+k1  [k1 ]2 [n] j=1 k 1

which yields Eq. (9.15) when c = 2 by the definition and straightforward cancelation. Finally, we turn to the case when c ≥ 3. By the definition we have ? Hn? [c, s] = Hn−1 [c, s] +

qn ? H [s]. [n]c n

By the induction assumption, we see that X qn X δ(s)Hn? [c, s] = Hn−1 [p] + c Hn [p]. [n] pΠc

p λ

(9.18)

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289 ≈

e {θ}c−2 , λ 1
(−1), λb). Then Set Π0 = (0, {1}c−3 , λ1 ⊕ 1, λb1 ; {0}c−2 , λ; 1 X X ≈ ≈ eb1 ; 1
p 1 , pb1 ] Hn−1 [p] = Hn−1 [2 ⊕ p1 , pb1 ; 1 + pe1 , p pΠ0

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pΠc

n−1 =

k1 n−1+k  1 k1 pΠ0 n>k1 >···>kr ≥1

X

X

≈ r q k1 (1+ep1 )+Q(1
p 1 ,k1 ) Y fj (p, k). (− sgn(p1 ))k1 [k1 ]2+|p1 | j=2

Plugging this into Eq. (9.18) and using Eq. (9.13), we obtain X ≈ ≈ eb1 ; 1
p 1 , pb1 ] δ(s)Hn? [c, s] = Hn [2 ⊕ p1 , pb1 ; 1 + pe1 , p pΠ0

+

qn X qn X ≈ e ; 1
pb1 ], Hn [p1 , pb1 ; p Hn [p] − 2 c [n] [n] 0 pΠ

p λ

which implies δ(s)Hn? [c, s] =

X pΠc

Hn [p] −

qn X qn X Hn [p] + c Hn [p], 2 [n] [n] 00 pΠ

(9.19)

p λ ≈

≈

e 1, {θ}c−3 , λ 1
(−1), λb). Exwhere Π00 = (0, {1}c−3 , λ1 ⊕ 1, λb1 ; {0}c−2 , λ; 1 panding the second sum from Eq. (9.19), we see that X X X ≈ ≈ e p eb1 ; w, pb1 ], Hn [p] = Hn [w, pb1 ; w, pΠ00

≈ pΠ000 (w;w; e w)(0,{1}c−3 ,p1 ⊕1; ≈ c−2 {0} ,e p1 ;1,{θ}c−3 ,p 1 )

≈

≈

e λ 1
(−1), λb). Applying Lemma 9.2.2 to the inner sum where Π000 = (λ; λ; 1 ≈ ≈ eb1 ; pb1 ), and c replaced by c − 2, with (a, b, r) = (p1 , pe1 , p 1 ), (x; y; z) = (pb1 ; p we finally arrive at X X X 1 1 ≈ ≈ e Hn [p] = H [p , p ; p ; p
1, p ] = Hn [p]. n 1 b b 1 1 1 [n]c−2 [n]c−2 00 000 pΠ

pΠ

p λ

(9.20) To justify the last equality above we need to show that for the components ≈

≈

≈

of λ satisfying Eq. (9.14) we have λ 1
(−1)
1 = λ 1 and for any j ≥ 2, ≈

≈

≈

≈

≈

≈

λ 1
(−1)
λ 2
· · ·
λ j
1 = λ 1
λ 2
· · ·
λ j .

These can be proved by using the projection π of Lemma 9.1.2 and the fact ≈

≈

≈

≈

that π(λ 1
(−1)
· · ·
λ j
1) = π(λ 1
· · ·
λ j ) ∈ {1, 2} by Eq. (9.14). Now by Eq. (9.19) and Eq. (9.20), we see that Eq. (9.15) is true for all c ≥ 3. We have completed the proof of the theorem.

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We now state the main q-MH? S identity that will lead to a few applications later. First, we need some extra notation. For any two compositions a = (a1 , . . . , am ) and b = (b1 , . . . , bn ), we define binary operations K K a b = (a1 , . . . , am−1 , am b1 , b2 , . . . , bn ) J where = ⊕, +,
for the three components of the triple mollifiers. For s ∈ N` , we define a sequence of admissible triple mollifiers mi (s) := ≈ e i , mi ) (i = `, . . . , 2, 1) (backward) inductively as follows. (mi , m   if si = 1;  (1, mi+1 ), (1), if s` = 1; m` = m = (2) ⊕ m , if si = 2; i i+1  (2, {1}s` −2 ), if s` ≥ 2, ¯ ⊕ mi+1 , if si ≥ 3, (2, {1}si −3 , 1)   e i+1 ), if si = 1;  (1, m (1), if s` = 1; e` = e i = (1) + m e i+1 , m m if si = 2;  (1, {0}s` −2 ), if s` ≥ 2, e i+1 ), (1, {0}si −3 , m if si ≥ 3,  ≈   if si = 1;  (2, −2)
mi+1 , (2), if s` = 1; ≈ ≈ ≈ mi = mi+1 , m` = if si = 2;  (1, {θ}s` −2 ), if s` ≥ 2, ≈  si −3 (1, {0} , −1)
mi+1 , if si ≥ 3, for all i < `.

Theorem 9.3.3. For all s = (s1 , . . . , s` ) ∈ N` , we have  X 1, if s` = 1; Hn? [s] = δ(s) Hn [p], where δ(s) = −1, if s` ≥ 2. p m1 (s)

Proof. The theorem follows from Thms. 9.3.1 and 9.3.2 easily. 9.4 Some Applications By repeatedly applying Thm. 9.3.3 we can now obtain a number of different types of identities. The following two results give us the q-analog of so-called 2-1-2 formulas (or simply the Two-one formulas). Here the underline means the string type ({2}a , 1) may be repeated an arbitrary number of times where a may vary in the repetitions. Corollary 9.4.1. Suppose ` ∈ N0 and a1 , . . . , a` ∈ N0 . Then X Hn? [{2}a1 , 1, . . . , {2}a` , 1] = Hn [p] pΠ

where Π =



 ` ` Cat{2aj + 1}; Cat{aj + 1}; 2, {0}`−1 . j=1

j=1

(9.21)

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Corollary 9.4.2. Suppose ` ∈ N0 , a1 , . . . , a` ∈ N0 and a`+1 ∈ N. Then

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Hn? [{2}a1 , 1, . . . , {2}a` , 1, {2}a`+1 ] = −

where Π =



X

Hn [p]

(9.22)

pΠ

 ` ` Cat{2aj + 1}, 2a`+1 ; Cat{aj + 1}, a`+1 ; 2, {0}`−1 , −1 . j=1

j=1

We can also get the following q-analog of the 2-c-2 (c ≥ 3) formula. Corollary 9.4.3. Suppose `, aj , cj ∈ N0 and cj ≥ 3 for all 1 ≤ j ≤ ` + 1. Set s = ({2}a1 , c1 , . . . , {2}a` , c` , {2}a`+1 ). Then Hn? [s] = −

X

Hn [p],

(9.23)

pΠ(s)

 `  where Π(s) = 2a1 + 2, {1}c1 −3 , Cat 2aj + 3, {1}cj −3 , 2a`+1 + 1; j=2  `  ` Cat aj + 1, {0}cj −3 , a`+1 ; 1, Cat{θ}cj −2 . j=1

j=1

Remark 9.4.4. If we take cj = 3 for all j then we obtain the q-analog of so-called Two-three formula. 9.5 Congruences of q-MHSs From last chapter we see that it is quite fruitful to study the van Hamme type congruence of the MHSs. In this section, abiding by the same principle we will consider some congruences of the q-MHSs of the same type. But this time we treat q-MHSs as rational functions in q. Note that [p] = [p]q is an irreducible polynomial in Z[q] when p is a prime. So we have the q-analog of Wolstenhomle’s Theorem Eq. (8.1) given below. Theorem 9.5.1. For any prime p ≥ 5, p−1 X 1 p−1 p2 − 1 ≡ (1 − q) + (1 − q)2 [p] [j] 2 24 q j=1 p−1 X 1 (p − 1)(p − 5) ≡− (1 − q)2 2 [j] 12 q j=1

(mod [p]2q ), (mod [p]).

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Or, equivalently, p−1 X

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j=1 p−1 X j=1

p − 1 p2 − 1 1 ≡ + (1 − q)[p] 1 − qj 2 24

(mod [p]2q ),

(9.24)

(mod [p]).

(9.25)

(p − 1)(p − 5) 1 ≡− j 2 (1 − q ) 12

Actually we will prove some generalizations of the above congruences. As in the previous sections we will suppress the dependence on q and write [j] = [j]q for all j ∈ N. For s = (s1 , . . . , sd ) ∈ Nd and t = (t1 , . . . , td ) ∈ Zd , we set (t) X q k1 t1 +···+kd td Hn [s] t . (9.26) , h [s] := Hnt [s] := n [k1 ]s1 · · · [kd ]sd (1 − q)|s| n≥k1 >···>kd ≥1

(s −1,...,s −1)

d [s] are the partial sums of the type I q-analogs Note that Hn 1 of the multiple zeta functions to be defined in Chap. 11. For trivial t, we set

Hn0 [s] := Hn(0,...,0) [s],

h0n [s] = Hn0 [s]/(1 − q)|s| .

To state the first main theorem of this section we define the degenerate Bernoulli numbers βn (λ) by the generating series ∞ X x xn βn (λ) . = 1/λ n! (1 + λx) − 1 n=0 Clearly we have limλ→0 βn (λ) = Bn . Remark 9.5.2. The degenerate Bernoulli numbers βn (λ) are polynomials of λ of degree n. More precisely, Howard proves in [301, Thm. 3.1] that β0 (λ) = 1, β1 (λ) = (λ − 1)/2 and [n/2]

βn (λ) = ˜bn λn + n

X B2j s(n − 1, 2j − 1)λn−2j 2j j=1

∀n ≥ 2,

(9.27)

where s(n, j) are the Stirling numbers of the first kind (cf. Eq. (5.37)) defined by n X x(x − 1)(x − 2) · · · (x − n + 1) = s(n, j)xj , j=0

and ˜bn are the Bernoulli numbers of the second kind defined by ∞ n 2 3 4 X x ˜bn x = 1 + 1 x − 1 x + 1 x − 19 x + · · · . = log(1 + x) n=0 n! 2 1! 6 2! 4 3! 24 4!

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Theorem 9.5.3. If p > 3 is a prime, then for all integers n > 1 we have (1)

hp−1 [n] =

p−1 X

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j=1

qj ≡ Kn (p) (1 − q j )n

(mod [p]),

where Kn (p) =

(−1)n+1 n  1  p βn n! p

= (−1)n+1

[n/2] ˜bn (−1)n X B2j − s(n − 1, 2j − 1)p2j . n! (n − 1)! j=1 2j

(9.28)

√ (1) Proof. Set fn (q) = hp−1 [n]q . Let µ = µp = exp(2π −1/p) be the primitive pth root of unity. Then as a polynomial in Z[q] we have [p]q =

p−1 Y

(q − µk ).

k=1

Hence, by the Remainder Theorem we only need to show that fn (µk ) = Kn (p) for all k = 1, . . . , p − 1. But fn (µk ) =

p−1 X j=1

p−1

X µjk µj = = fn (µ). (1 − µjk )n (1 − µj )n j=1

Now one can check that (see Exercise 9.6) p−1 p−1 X X µj µj 1 1 p − = = z−1 p j j 1−z 1−z µ − z µ − 1 1 − µj −1 j=1 j=1

=

p−1 X j=1

µj

(9.29)

∞ X

∞ X (z − 1)n = (−1)n+1 fn+1 (µ)(z − 1)n . j − 1)n+1 (µ n=0 n=0

On the other hand, by definition ∞ X

(−1)n+1 Kn+1 (p)(z − 1)n =

n=0


n+1 1 X βn+1 (1/p) p(z − 1) 1−z (n + 1)! n≥0

p 1 = − . 1 − zp 1−z

We now finish the proof of the theorem by comparing the above two generating functions. Note that the second equation in Eq. (9.28) follows from Remark 9.5.2.

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Corollary 9.5.4. If p > 3 is a prime, then for every integer n ≥ 0 we have (0)

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hp−1 [n] ≡ p +

n X

Kj (p)

(mod [p]).

j=0

(0)

Proof. If n = 0 then hp−1 [n] = p − 1 while K0 (p) = −1. Thus the corollary is true. Suppose n ≥ 1. Then we have p−1 X

(0)

hp−1 [n] =

j=1

p−1

X 1 − qj + qj 1 (0) = ≡ hp−1 [n−1]+Kn (p) j n j )n (1 − q ) (1 − q j=1

(mod [p]).

Hence the corollary follows by an easy induction. Corollary 9.5.5. If p > 3 is a prime, then for every integer n > t ≥ 1 we have  p−1 t−1  X X t−1 q tj (t) (−1)i Kn−i (p) (mod [p]). (9.30) ≡ hp−1 [n] = j )n i (1 − q i=0 j=1 Moreover, p−1 X j=1

  n X 1 p − 1 Kj (p) ≡ + (1 − q j )n 2 j=2

(mod [p]).

(9.31)

Proof. If t > 1 it is clear that q

tj

 t−1  X
t−1 t−1 j = q 1 − (1 − q ) =q (−1)i (1 − q j )i . i i=0 j

j

So Eq. (9.30) follows from Thm. 9.5.3 while Eq. (9.31) follows from Cor. 9.5.4 and Eq. (9.27). √ Theorem 9.5.6. Let s be a positive integer and µs = exp(2π −1/s) the primitive sth root of unity. Then ∞ X d=0

s−1    (−1)s Y  h0p−1 {s}d xd ≡ s 1 − (1 − µns (−x)1/s )p (mod [p]). p x n=0

√ Proof. Let µ = exp(2π −1/p) be the primitive pth root of unity and set Pn =

p−1 X j=1

1 − pδ0,n (1 − µj )n

∀n ≥ 0,

(9.32)

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where δ0,n is the Kronecker symbol. Similar to the argument in the proof of Thm. 9.5.3 we see that h0 (n) ≡ Pn (mod [p]) for all n ≥ 1. By partial fraction expansion in Eq. (9.29): g(x) :=

∞ X

p−1 X

n

Pn x =

n=0 p−1 X

1 − µj −p 1 − x − µj

j=1 −j

p−1 X

µj − p (z = 1 − x) zµ−j − 1 j=1 z − µj j=1     p 1 p 1 −1 = −z − − − −p z −p − 1 z −1 − 1 zp − 1 z − 1 px(x − 1)p−1 =− . 1 − (1 − x)p   P∞ Let ad = h0p−1 {s}d for all d ≥ 0. Let w(x) = d=0 ad xd be its generating function. By =

µ

−

d−1       1X (−1)d−k−1 h0p−1 (d − k)s · h0p−1 {s}k h0p−1 {s}d = d

(9.33)

k=0

(see Exercise 9.5), we get w(x) =

∞ X d=0

ad xd ≡ 1 +

∞ d−1 X 1X d=1

d

(−1)d−k−1 P(d−k)s ak xd

(mod [p]).

k=0

Differentiating both sides and changing index d → d + 1, we get, modulo [p], w0 (x) ≡

∞ X d X

(−1)d−k P(d−k+1)s ak xd ≡

d=0 k=0

∞ X ∞ X

(−1)d−k P(d−k+1)s ak xd .

k=0 d=k

By d → d + k and then changing the order of summation, we get ! ∞ ∞ X w(x) X d 0 d P(d+1)s (−x) ≡ w (x) ≡w(x) Pds (−x) + 1 −x d=0 d=0 ! ∞ s−1 X X w(x) n 1/s d Pd (µs (−x) ) ≡ s+ −sx n=0 d=0

≡

w(x)  −sx

w(x) ≡ −sx

s+

s−1 X

g µns (−x)1/s




n=0 s−1 X pµn (−x)1/s (µn (−x)1/s − 1)p−1 s− 1 − (1 − µns (−x)1/s )p n=0

! .

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Thus 0

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(ln w(x)) =

s−1 X (1 − (1 − µn (−x)1/s )p )0 −(ln x) + 1 − (1 − µns (−x)1/s )p n=0

!

0

.

Therefore by comparing the constant term we get w(x) ≡

s−1  (−1)s Y  (mod [p]) 1 − (1 − µns (−x)1/s )p s p x n=0

as desired. Corollary 9.5.7. For every positive integer d < p, we have    d 1 p−1 0 hp−1 {1} ≡ (mod [p]q ). d+1 d Proof. By the theorem we get ∞ X

h0p−1

d=0

 ∞   d d p 1 X (1 + x)p − 1 ≡ xd+1 {1} x ≡ d+1 px px d=0   ∞ X p−1 d 1 x ≡ d+1 d

(mod [p]q ).

d=0

The corollary follows immediately. Corollary 9.5.8. For every positive integer d < p, we have         X   (−1)d  X p p  p p   h0p−1 {2}d ≡ − p2  2j + 1 2k + 1 2j 2k  j+k=d 0≤j,kk` ≥1

.

Let k = (k1 , . . . , k` ) ∈ N` and O(k) = {σ ∈ S` : nσ(j) = nj ∀j} its associated isotropy subgroup. Let Π(k) be the corresponding partition of {1, . . . , `}. A term like sgn(n1 )kσ(1) · · · sgn(n` )kσ(`) |n |

|n |

1 ` kσ(1) · · · kσ(`)

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327

appears exactly once on the left-hand side of Eq. (10.33) if all kj are distinct. Otherwise, it does not appear at all. We now only need to show that for each fixed k,  q X Y 1, if ]Π(k) = k; `−q (−1) (]πh − 1)! = (10.34) 0, otherwise. (π1 ,...,πq )Π(k)

h=1

Indeed, first note that (−1)`−q = 1 if and only if the permutations of cycle type (π1 , . . . , πq ) are even (check this!). Second, for each πh of eh elements, there corresponds (eh − 1)! different cycles of length eh . So the left-hand side of Eq. (10.34) is the signed sum of the number of even and odd permutations in O(k). This number is clearly 0 unless Π(k) is the trivial partition {{1}, . . . , {`}} in which case the number becomes 1. This concludes the proof of the theorem. Corollary 10.9.2. Let n1 , . . . , n` ∈ D. If nj are all even then X
ζ nσ(1) , . . . , nσ(`) ∈ π |n1 |+···+|n` | Q. σ∈S`

Proof. Since ζ(2n) = (21−2n − 1)ζ(2n) ∈ π 2n Q for every positive integer n the corollary follows from Lemma 10.9.1 immediately . Theorem 10.9.3. Let r be a positive integer, and e1 , . . . , e2r+1 nonnegative integers. (i) Put m = e1 + · · · + e2r . Then the sum X ζ ? ({2}eτ (1) , 3, {2}eτ (2) , 1, {2}eτ (3) , . . . , 3, {2}eτ (2r) , 1) τ ∈S2r

is a rational multiple of π 2m+4r . (ii) Put m = e1 + · · · + e2r+1 . Then the sum X ζ ? ({2}eτ (1) , 3, {2}eτ (2) , 1, . . . , 3, {2}eτ (2n) , 1, {2}eτ (2r+1) +1 ) τ ∈S2r+1

is a rational multiple of π 2m+4r+2 . Proof. We start with (i) first. When r = 1 this follows quickly from Eq. (10.31) by the stuffle relation. For general r, let aj = e2j and bj = e2j−1 for all j ≤ r and let Aj = 2ej + 2 for all j ≤ 2r. Then we can apply Eq. (10.30) of Prop. 10.8.8 to the string s = ({2}b1 , 3, {2}a1 , 1, . . . , {2}b` , 3, {2}a` , 1) and get X ζ ? (s) = ζ ] (A1 , . . . , A2r ) = 2`(p) ζ(p). p(A1 ,...,A2r )

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For s = ({2}e1 , 3, {2}e2 , 1, . . . , {2}e2r−1 , 3, {2}e2r , 1) and any permutation τ ∈ S2r , we define Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 06/12/16. For personal use only.

sτ = ({2}eτ (1) , 3, {2}eτ (2) , 1, . . . , {2}eτ (2r−1) , 3, {2}eτ (2r) , 1) Let A = (A1 , . . . , A2r ), Aτ = (Aτ (1) , . . . , Aτ (2r) ), and P` (2r) be the set of all partitions of [2r] := {1, 2, . . . , 2r} into ` consecutive subsets. If λ = (λ1 , . . . , λ` ) ∈ P` (2r) then we set λj (Aτ ) = (Aτ (i) )i∈λj so that the F` concatenation j=1 λj (Aτ ) = Aτ . Because of the permutation we see that X

ζ ? (sτ ) =

2r X X τ ∈S2r `=1

τ ∈S2r

=

X

2r X

τ ∈S2r `=1

X

2`

λ∈P` (2r)

2` `!

X

ζ ⊕ λ1 (Aτ ), . . . , ⊕λ` (Aτ ) X

λ∈P` (2r) g∈S`





ζ ⊕ λg(1) (Aτ ), . . . , ⊕λg(`) (Aτ ) ,

where ⊕t is the ⊕-sum of all the components of t for any composition t. Hence Thm. 10.9.3(i) follows readily from the Cor. 10.9.2 since all Aj ’s are even numbers. Thm. 10.9.3(ii) follows from Eq. (10.48) in Exercise 10.8 in a similar fashion, so we leave the details to the interested reader. Theorem 10.9.4. For any positive integers n and m, we have ζ ? ({2}m , 1)ζ ? ({2}n , 1) = ζ ? ({2}m , 1, {2}n , 1)+ζ ? ({2}n , 1, {2}m , 1), (10.35) ζ ? ({2}m , 1)ζ ? ({2}n ) = ζ ? ({2}m , 1, {2}n )+ζ ? ({2}n−1 , 3, {2}m ), ?

m

?

n

?

ζ ({2} )ζ ({2} ) = ζ ({2}

m−1

n−1

, 3, {2}

?

n−1

, 1)+ζ ({2}

(10.36) m−1

, 3, {2}

, 1). (10.37)

Proof. By Prop. 10.8.3 and Eq. (10.28) we have 1 ? 1 ζ ({2}m , 1, {2}n , 1) = ζ(2m + 1, 2n + 1) + ζ(2m + 2n + 2), 4 2 1 ? ζ ({2}m , 1) · ζ ? ({2}n , 1) = ζ(2m + 1)ζ(2n + 1) 4 = ζ(2m + 1, 2n + 1)+ ζ(2m + 2n + 2) + ζ(2n + 1, 2m + 1), 1 1 − ζ ? ({2}m , 1, {2}n ) = ζ(2m + 1, 2n) + ζ(2m + 2n + 1), 4 2 1 ? m ? n − ζ ({2} , 1) · ζ ({2} ) = ζ(2m + 1)ζ(2n) 4 = ζ(2m + 1, 2n)+ ζ(2m + 2n + 1) + ζ(2n, 2m + 1),

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1 ? ζ ({2}m ) · ζ ? ({2}n ) = ζ(2m)ζ(2n) 4 = ζ(2m, 2n) + ζ(2m + 2n) + ζ(2n, 2m), by the stuffle relations. On the other hand, by Eq. (10.31) and Eq. (10.29), we get 1 ? 1 ζ ({2}n−1 , 3, {2}m−1 , 1) = ζ(2m + 2n) + ζ(2n, 2m), 4 2 1 ? 1 m−1 n−1 ζ ({2} , 3, {2} , 1) = ζ(2m + 2n) + ζ(2m, 2n), 4 2 1 1 ? n−1 m − ζ ({2} , 3, {2} ) = ζ(2n, 2m + 1) + ζ(2m + 2n + 1). 4 2 The rest of the proof is straightforward and is thus left to the interested reader. Theorem 10.9.5. Let m and n be two nonnegative integers. (i) We have (2n + 1)ζ ? ({3, 1}n , 2) =

X

ζ ? ({3, 1}j )ζ ? ({2}2k+1 ).

j+k=n

(ii) If n ≥ 1 then we have X ζ ? ({2}e1 , 3, {2}e2 , 1, . . . , {2}e2n−1 , 3, {2}e2n , 1) e1 +e2 +···+e2n =1 e1 ,e2 ,...,e2n ≥0

=

X

ζ ? ({3, 1}j , 2)ζ ? ({2}2k+2 ).

j+k=n−1

Remark 10.9.6. Note that Thm. 10.9.5(ii) is related to Exercise 10.10 but is not a consequence of it. Also notice there is no 2-string at the end of any MZ? V on the left-hand of (ii). Further, the statement in Thm. 10.9.5 is generally false if ζ ? is replaced by ζ. Proof. (i) By removing all the 2’s in 2-3-2-1 formula in Prop. 10.8.8 we get X
ζ ? ({3, 1}j ) = ζ ] {2}2j = 2`(p2j ) ζ(p2j ).
p2j  {2}2j

All of the components aj of {2 ◦ · · · ◦ 2} must satisfy the following sign rule: aj > 0 if and only if 4|aj .

(10.38)

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On the other hand, by removing all the 2’s in 2-c-2-1-2 formula (see Exercise 10.8) except the last 2, we have X ζ ? ({3, 1}n , 2) = − 2`(p2n+1 ) ζ(p2n+1 ).
p2n+1  {2}2n+1

Hence by Eq. (10.28) we need to show that X 2`(p2n+1 ) ζ(p2n+1 ) (2n + 1)
p2n+1  {2}2n+1

=

n X j=0

X p2j 

{2}2j




2`(p2j ) ζ(p2j ) · 2ζ(4(n − j) + 2).

(10.39)


Suppose a composition p2n+1  {2}2n+1 has the form p2n+1 = (a1 , . . . , at ) ∈ Dt , t

1 ≤ t ≤ 2n + 1.

We now show that there are exactly 2 (2n+1) copies of such term produced by the stuffle product on the right-hand side of Eq. (10.39). Indeed, for each i = 1, . . . , t the entry ai has two cases: Case (1). ai = 4bi > 0. Then for each k = 1, . . . , bi we may produce such a term on the right-hand side of Eq. (10.39) by stuffing 2t ζ(a1 , . . . , ai−1 , 4k − 2, ai+1 , . . . , at ) from p2j having length t with the term 2ζ(4(bi − k) + 2) at the right end of Eq. (10.39). Notice no shuffle is possible since 4(n − j) + 2 is not a multiple of 4. Hence these contribute to 2t+1 bi = 2t−1 ai copies of ζ(a1 , . . . , at ). Case (2). ai = 4bi + 2. Then for each k = 1, . . . , bi we may produce such a term on the right-hand side of Eq. (10.39) by stuffing 2t ζ(a1 , . . . , ai−1 , 4k, ai+1 , . . . , at ) from p2j having length t with the term 2ζ(4(bi − k) + 2) at the right end of Eq. (10.39). Further, there is exactly one possible shuffle given by  2t−1 ζ(a1 , . . . , ai−1 , ai+1 , . . . , at+1 ) 2ζ(4bi + 2) ,




since the index (a1 , . . . , ai−1 , ai+1 , . . . , at ) has only length t − 1. Altogether these produce 2t+1 bi + 2t = 2t−1 |ai | copies of ζ(a1 , . . . , at ). By combining Case (1) and Case (2), we see that the right-hand side of Eq. (10.39) produces exactly t X i=1

2t−1 |ai | = 2t−1 · |(a1 , . . . , at )| = 2t (2n + 1)

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copies of ζ(a1 , . . . , at ) since the weight is 4n + 2. This proves (i).

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(ii) We use the same analysis as above and see that we need to prove the following identity: X q2n

2`(q2n ) ζ(q2n ) =

n−1 X j=0

X p2j+1  {2}2j+1




2`(p2j+1 ) ζ(p2j+1 ) · 2ζ(4(n − j)),

(10.40) where q2n runs through all indices of the form A1 ◦ · · · ◦ A2n with one of the Aj ’s (say Aj0 ) equal to 4 and all the other Aj ’s equal to 2. For each choice of 2t ζ(a1 , . . . , at+1 ) with length t + 1 from the left-hand side of Eq. (10.40), all but one of the argument components a1 , . . . , at+1 must satisfy the sign rule Eq. (10.38). The only exceptional component, say ai , must involve a merge with the special entry Aj0 = 4. Now there are two cases: Case (1). ai = 4bi + 2 > 0. Then for each k = 0, . . . , bi − 1 we may produce such a term on the right-hand side of Eq. (10.40) by stuffing 2t ζ(a1 , . . . , ai−1 , 4k + 2, ai+1 , . . . , at+1 ) from p2j+1 having length t + 1 with the term 2ζ(4(bi − k)) at the right end of (10.40). Notice no shuffle is possible since 4(n − j) is a multiple of 4. Hence these contribute to 2t+1 bi copies of ζ(a1 , . . . , at+1 ). On the left-hand side, such a term must be produced by setting all 2bi − 1 consecutive ◦’s around Aj0 = 4 to ⊕: . . . , Ai ⊕ Ai+1 ⊕ · · · ⊕ Aj0 ⊕ · · · ⊕ A` , . . . . {z } | 2bi entries

But Aj0 can be at any one of the 2bi possible positions, thus producing 2t+1 bi copies of ζ(a1 , . . . , at+1 ) which match exactly the right-hand side of (10.40). Case (2). ai = 4bi . Then for each k = 1, . . . , bi − 1 we may produce such a term on the right-hand side of (10.40) by stuffing 2t ζ(a1 , . . . , ai−1 , 4k, ai+1 , . . . , at+1 ) from p2j having length t + 1 with the term 2ζ(4(bi − k)) at the right end of Eq. (10.40). Further, there is exactly one possible shuffle given by  2t−1 ζ(a1 , . . . , ai−1 , ai+1 , . . . , at+1 ) 2ζ(4bi ) .




t+1

t

Hence these contribute to 2 (bi − 1) + 2 = 2t (2bi − 1) copies of ζ(a1 , . . . , at+1 ). Similar to (1), on the left-hand side, such a term must be produced by setting all 2bi − 2 consecutive ◦’s around Aj0 to ⊕. And

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Aj0 can be at any one of the 2bi − 1 possible positions, thus producing 2t (2bi − 1) copies of ζ(a1 , . . . , at+1 ) which match exactly the right-hand side of Eq. (10.40). This concludes the proof of theorem. Next we will prove a more general formula similar to Thm. 10.9.4. First, we introduce some more notation. For ∞ ≥ A ≥ B ≥ 1, we put S−1 (A, B) = δA,B A2 , S0 (A, B) = 1, X 1 (j = 1, 2, 3, . . .), Sj (A, B) = a21 · · · a2j A≥a1 ≥···≥aj ≥B

where δA,B is the Kronecker symbol. Then we have A A X X 1 1 Sj (A, B) = S (p, B) = Sj−1 (A, p) 2 j−1 p2 p p=B

(10.41)

p=B

for any j ≥ 0. We now prove a lemma by using partial fractions. Lemma 10.9.7. For any j ≥ −1 and 1 ≤ p, q ≤ ∞, we have p q X X q p Sj (p, p0 ) = Sj (q, q0 ) . p0 (p0 + q) q =1 q0 (q0 + p) p =1 0

(10.42)

0

Here, p0 (pq0 +q) (resp. and so on.

p q0 (q0 +p) )

means

1 p0

(resp.

1 q0 )

if q = ∞ (resp. p = ∞),

pq Proof. We use induction on j. When j = −1, both sides are equal to p+q . Thus we assume j ≥ 0. By Eq. (10.41), we can rewrite the left-hand side of Eq. (10.42) as   p p p X X X 1 1 q 1 = Sj−1 (p, p1 ) 2 Sj (p, p0 ) − p0 (p0 + q) p =1 p =p p1 p0 p0 + q p0 =1 1 0 0   p p 1 X 1 X 1 1 = Sj−1 (p, p1 ) 2 − . p1 p =1 p0 p0 + q p =1 1

0

By using the identity  p1  X 1 1 − p0 p0 + q p0 =1     ∞ X 1 1 1 1 = − − − p0 p0 + q p0 + p1 p0 + p1 + q p0 =1  X q  q X 1 1 p1 = − = , p p + p q (q + p1 ) 0 0 1 1 1 p =1 q =1 0

1

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we get  p p1  q p X X Sj−1 (p, p1 ) X 1 p1 1 Sj−1 (p, p1 ) X − = 2 2 p p p + q p q (q + p1 ) 0 0 1 1 p =1 p =1 q =1 1 1 p =1 Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 06/12/16. For personal use only.

1

0

1

1

q p X 1 X q1 Sj−1 (p, p1 ) = . q2 p1 (p1 + q1 ) q =1 1 p =1 1

1

Finally, using the induction hypothesis and Eq. (10.41) again, we obtain p q1 q q X X 1 X pSj−1 (q1 , q0 ) 1 X q1 Sj−1 (p, p1 ) = q2 p1 (p1 + q1 ) q2 q0 (q0 + p) q =1 1 q =1 q =1 1 p =1 1

1

0

1

=

q X q0

pSj (q, q0 ) . q (q + p) =1 0 0

Thus we have shown Eq. (10.42). For n ∈ N, let j = (j1 , . . . , jn ) ∈ (N0 )n and e = (e1 , . . . , en ) where ej = 1 or 3. Define
C(j; e) := ζ ? {2}j1 , e1 , {2}j2 , e2 , . . . , en−1 , {2}jn . (10.43) For n = 0, we simply put C = 1. Note that the right-hand side of Eq. (10.43) diverges if and only if n ≥ 2, j1 = 0 and e1 = 1. Otherwise, (j, e) is called an admissible pair. For convenience, we put ja↑b = (ja , ja+1 , . . . , jb ) and jb↓a = (jb , jb−1 , . . . , ja ) for any a ≤ b. Theorem 10.9.8. Let n be a positive integer, j = (j1 , . . . , jn ) ∈ (N0 )n and e = (e1 , . . . , en ) with ej = 1 or 3 for all j. Assume that both (j, e) and ← ←

( j , e) are admissible pairs. For k = 0, . . . , n, if ek = 1 then we define X(k) := C(j1↑k , 0; e1↑k ) · C(jn↓(k+1) ; e(n−1)↓k ) and if ek = 3 then we define X(k) := C(j1↑(k−1) , jk + 1; e1↑(k−1) ) · C(jn↓(k+2) , jk+1 + 1; e(n−1)↓(k+1) ). Then we have n X k=0

(−1)k X(k) = 0.

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Proof. Set e0 = en = 1. For k = 0, . . . , n, put X

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E(k) =

p0 ≥···≥pk ≥1 qn+1 ≥qn ≥···≥qk+1 ≥1

X

F (k) =

p0 ≥···≥pk ≥1 qn+1 ≥qn ≥···≥qk+1 ≥1

n k Y Sjα (pα−1 , pα ) Y Sjβ (qβ+1 , qβ ) pkek −1 qk+1 · , e peαα pk + qk+1 qββ−1 α=1 β=k+1

k n ek −1 Y Sjα (pα−1 , pα ) Y Sjβ (qβ+1 , qβ ) pk qk+1 · . e peαα pk + qk+1 qββ−1 α=1 β=k+1

Here we allow p0 = ∞ and qn+1 = ∞. In particular, we have E(0) = 0 q1 n (resp. pnp+∞ ). For general k, (resp. F (n) = 0) because of the factor ∞+q 1 we see that E(k) + F (k) = X(k) since  ek −1 pk qk+1 pekk −1 qk+1 1, if ek = 1; + = pk qk+1 , if ek = 3. pk + qk+1 pk + qk+1 Moreover, when 1 ≤ k ≤ n, one has pk−1

qk+1 X Sj (pk−1 , pk ) pek −1 qk+1 X Sj (qk+1 , qk ) pk−1 q ek−1 −1 k k k k = ek−1 ek p p + q p + qk q k k+1 k−1 k k p =1 q =1 k

k

by Lemma 10.9.7, hence E(k) = F (k − 1). Therefore, n X

(−1)k X(k) =

k=0

n X

(−1)k E(k) + F (k)




k=0

=

n X

(−1)k F (k − 1) +

k=1

n−1 X

(−1)k F (k) = 0

k=0

as desired. Example 10.9.9. Taking e = (3, 3) or e = (3, 1) in Thm. 10.9.8, one has ζ ? ({2}j1 , 3, {2}j2 , 3, {2}j3 , 1) + ζ ? ({2}j1 +1 ) · ζ ? ({2}j3 , 3, {2}j2 +1 )

= ζ ? ({2}j1 , 3, {2}j2 +1 ) · ζ ? ({2}j3 +1 ) + ζ ? ({2}j3 , 3, {2}j2 , 3, {2}j1 , 1)

or ζ ? ({2}j1 , 3, {2}j2 , 1, {2}j3 , 1) + ζ ? ({2}j1 +1 ) · ζ ? ({2}j3 , 1, {2}j2 +1 )

= ζ ? ({2}j1 , 3, {2}j2 , 1) · ζ ? ({2}j3 , 1) + ζ ? ({2}j3 , 1, {2}j2 , 3, {2}j1 , 1),

respectively. Of course, these two relations can by proved by applying Thm. 10.8.1 to each of the MZ? Vs and then using the stuffle relations. Corollary 10.9.10. Let n > 0 be an integer. Then

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(i) For any a1 , a2 , . . . , an ∈ N0 such that a1 , an ≥ 1, we have n X

(−1)k ζ ?



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k=0

Here ζ ?



 k  k+1  {2}aj , 1 j=1 · ζ ? {2}aj , 1 j=n = 0.

(10.44)

 0  n+1  {2}aj , 1 j=1 = ζ ? {2}aj , 1 j=n = 1.

(ii) For any a1 , a2 , . . . , a2n ∈ N0 , we have n X k=0

ζ ? ({2}a1 , 3, {2}a2 , 1, . . . , 3, {2}a2k , 1) × ζ ? ({2}a2n , 3, {2}a2n−1 , 1, . . . , 3, {2}a2k+1 , 1) n X = ζ ? ({2}a1 , 3, {2}a2 , 1, . . . , 1, {2}a2k−1 +1 )

(10.45)

k=1 ?

× ζ ({2}a2n , 3, {2}a2n−1 , 1, . . . , 1, {2}a2k +1 ). Proof. Taking e = ({1}n ) we get

X(k) = C j1↑k , 0; {1}k · C jn↓(k+1) , 0; {1}n−k

= ζ ? ({2}j1 , 1, . . . , {2}jk , 1) · ζ ? ({2}jn , 1, . . . , {2}jk+1 , 1).

Hence Thm. 10.9.8 implies the identity Eq. (10.44). Similarly, for e = ({3, 1}n ), we have X(k) =ζ ? ({2}j1 , 3, {2}j2 , 1, . . . , 3, {2}jk , 1)

× ζ ? ({2}j2n , 3, {2}j2n−1 , 1, . . . , 3, {2}jk+1 , 1)

if k is even, and X(k) =ζ ? ({2}j1 , 3, {2}j2 , 1, . . . , 3, {2}jk−1 , 1, {2}jk +1 )

× ζ ? ({2}j2n , 3, {2}j2n−1 , 1, . . . , 3, {2}jk+2 , 1, {2}jk+1 +1 )

if k is odd. This proves Eq. (10.45). Proposition 10.9.11. For positive integers m and n, we have ?

ζ {2m}

n




= π

2mn

X Pm−1

(−1)

nj =mn ni ≥0

j=0

where β(n) = (2 − 22n )B2n /(2n)!.

mn

exp

m−1 2πi X lnl m l=0

! m−1 Y k=0

β(nk )

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Proof. Using the infinite product for the sine function, we have !−1 ∞  Y πi πi x2 2πi k k k csc πxe m = πxe m 1 − 2e m . n n=1 Substituting k = 0, 1, . . . , m − 1 and multiplying both sides, we obtain ! m−1 −1 m−1 ∞  Y Y πi Y πi x2m k k m m m m . (10.46) csc πxe = π x e 1 − 2m n n=1 k=0

k=0

The right-hand side of Eq. (10.46) equals ! ∞ X 1 X 2m 1+ x + · · · = 1 + ζ ? ({2m}n )x2mn . 2m n n >0 1 n=1 1

On the other hand, the left-hand side of Eq. (10.46) equals ! m−1 ∞ m−1 Y πi Y X (22nk − 2)B2nk m m k π x em (−1)nk −1 (2nk )! n =0 k=0

×π =

k=0

2nk −1 2nk −1

x

m−1 Y

∞ X

(22nk − 2)B2nk 2nk 2nk 2πi knk π x em (2nk )!   m−1 m−1 X Y 2πi (−1)m(n−1) (πx)2mn exp  jnj  β(nj ). m j=0 j=0

(−1)nk −1

k=0 nk =0

=1+

e

k

πi m k(2nk −1)

∞ X

X

n=1 m−1 Σ nj =mn j=0

Comparing coefficients of both sides, we obtain the desired identity. 10.10 Historical Notes Even at the early stage of Euler’s study of the MZVs he in fact considered not the MZVs but the MZ? Vs. Notice sometimes these values were also called non-strict MZVs in some older literature (for e.g., [445]). Instead of Eq. (1.23) Euler actually showed, for all n ≥ 2 2ζ ? (n, 1) = (n + 2)ζ(n + 1) −

n−2 X i=1

ζ(n − i)ζ(i + 1),

where ζ ? (n, 1) is a double zeta star value. See [210]. In [288] Hoffman established a few fundamental results concerning both the MZVs and the MZ? Vs. Since then the MZ? Vs frequently appear in the literature although they do not seem to enjoy the same fame as their

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cousins, the MZVs. Most often one can reduce the study of the MZ? Vs to that of the MZVs but sometimes the MZ? V version is nicer and easier while at other times the opposite is true. To make things both more complicated and more interesting, the symmetrized MZVs (see Chap. 6) are even more elegant than both the MZVs and the MZ? Vs in some situations. The integral expression of the MZ? Vs in Thm. 10.1.1 was first stated in Yamamoto’s paper [583]. Using this idea, he gave a few very nice relations between the MZ? Vs and Arakawa–Kaneko zeta values (see [21]). In [288] Hoffman applied some combinatorial theory to the MZ? V and proved Thm. 10.2.2. Machide [410] then generalized the MZV part of this theorem to regularized MZVs of weight ≤ 4. The sum relation stated as Eq. (10.7) was first conjectured by C. Moen in an unpublished manuscript according to Hoffman [288]. It was initially verified for depth up to three by Moen and by Hoffman for weight up to six. Our proof here closely follows the argument of [288]. Ohno and Wakabayashi [446] proved the cyclic sum relation of Thm. 10.4.2 for the MZ? V by using partial fractions. In [534], Tanaka and Wakabayashi used Hoffman’s algebraic machinery to prove some more results akin to the cyclic sum formula. Soon afterwards, Ohno et al. [449] extended the cyclic sum formula to its q-analog. In [14] Aoki and Ohno used generating functions to derive the sum formula of the MZ? Vs of fixed weight and height, which is the content of Cor. 10.6.5. Their method was improved to handle the case with the depth also fixed in [15]. This is the source of Thm. 10.6.2. The identity Eq. (10.10) was proved by Zlobin [641]. He also mentioned it was essentially known to Vasilev [559]. Corollary 10.7.6 was first shown by Kaneko and Ohno [332, Thm. 1.3]. Another proof was given by Yamazaki [584]. It was later generalized to arbitrary height by Z. Li [397] whose proof is reproduced here to show Thm. 10.7.3. Proposition 10.8.3 was discovered by Ohno and Zudilin [448] numerically first. They dubbed the identity the “Two-one formula”, which we denote by the 2-1 formula here. In [582] Yamamoto considered some algebraic structures depending on a variable t which reflect the properties of the MZVs and the MZ? Vs when t = 0 and t = 1, respectively. As he pointed out (see [581, Conjecture 4.4]) the validity of the 2-1 formula implies that the MZ? V algebra structure of the form ζ ? ({2}a1 , 1, . . . , {2}a` , 1) is reflected by setting t = 1/2. As special cases of the “Two-one formula”, Eq. (10.28) (resp. Eq. (10.35)) was first proved by Zlobin [639] (resp. by Ohno and Zudilin [448] by partial fractions). In [241] Glanois studied Conjecture 10.8.7 using the motivic version of

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the general 2-3-2 formula and the Hopf algebra structure of motivic MZVs and Euler sums. She also obtained many nice results on the colored MZVs (see Chap. 13) using Galois descents. When all the arguments n1 , . . . , n` are positive Lemma 10.9.1 becomes [288, Thm. 2.2] whose proof is adopted here. Note that the same result is also stated as [368, Prop. 9.4] whose proof is different from that of [288] but also works here. The “Three-One” type formula in Theorem 10.9.8 is due to Tasaka and Yamamoto [537]. The “Two-three” type formula in Thm. 10.9.3 was first conjectured by Imatomi et al. as Conjectures 4.1 and 4.3 of [313]. Some special cases were confirmed by Imatomi et al. [313, Thm. 1.1] and also by Tasaka and Yamamoto [537]. Later, Yamamoto [581] presented a more precise version. Here we have given a different and concise proof. In [581, Thm. 1.1] Yamamoto obtained a more precise formula by using partial sums and generating functions: X ζ ? ({2}e0 , 3, {2}e1 , 1, {2}e2 , 3, . . . , 3, {2}e2r−1 , 1, {2}e2r ) e0 ,e1 ,...,e2r ≥0 e0 +e1 +···+e2r =m

X

=

(−1)j+k

2i+k+u=2r j+l+v=m



k+l k



u+v u



 0 0 π 4r+2m βk+l βu+v 2i + j , (2i + 1)(4i + 2j + 1)! j (10.47)

βn0

n

2n

where = (−1) (2 − 2 )B2n /(2n)!. It is possible to modify the proof of Thm. 10.9.3 to give this more quantified version. The “Three-Two-One” type formulas in Thm. 10.9.5 were first conjectured by Imatomi et al. as [313, Conjecture 4.5]. Note that Thm. 10.9.5(i) is the more precise version of the n = 1 case of Thm. 10.9.3(i). And Thm. 10.9.5(iii) can be written more compactly as n
X ζ ? {2} {3, 1}n = ζ ? ({3, 1}n−k , 2)ζ ? ({2}2k ),




k=0

which is the more precise version of the m = 1 case of Exercise 10.10, a result of Kondo et al. [358]. The case m = 0 case has the following precise formulation by Muneta [429]:  n  X X 2 ζ ? {3, 1}n ) = (−1)n1 β(n0 )β(n1 ) π 4n , (4i + 2)! i=0 n0 +n1 =2(n−i) n0 ,n1 ≥0

where β(n) = (2 − 22n )B2n /(2n)!. Muneta also found a precise form in the case m = 1 and Prop. 10.9.11. Of course, these are all special cases of

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Yamamoto’s general formula in Eq. (10.47). Finally, Tasaka and Yamamoto proved Thm. 10.9.4 in [537] while Kondo et al. found an MZ? V analog of Thm. 5.8.1 in [358]. Exercises 10.1. Suppose d, k1 , . . . , kd ∈ N. For any nonnegative integer i ≤ k1 − 2, prove that X

1

n1 ≥n2 ≥···≥nd ≥nd+1 ≥1, n1 6=nd+1

(n1 − nd+1 )n1k1 −i . . . nkdd nid+1

=

X

1

n1 ≥n2 ≥···≥nd ≥nd+1 ≥1, n1 6=nd+1

(n1 − nd+1 )nk11 −i−1 . . . nkdd ni+1 d+1

− ζ ? (k1 − i, k2 , k3 , . . . , kd , i + 1) + ζ(|k| + 1) by mimicking the proof of Lemma 5.2.2. 10.2. Show that the cyclic sum relation Eq. (10.9) implies the sum relation Eq. (10.7). 10.3. Prove Eq. (10.14) by using geometric series. 10.4. Use Lemma C.1 to verify that the two functions in Eq. (10.15) are indeed the solutions of the corresponding homogeneous differential equation (10.13). Then verify that Eq. (10.16) gives a solution of Eq. (10.13). Note that the homogeneous equation is a special case of the Riemann differential equations with the singularities at 0, 1, ∞ (see Sec. C.1). 10.5. Prove the transformation Eq. (10.17) using the linear transformation Eq. (C.5) with (a, b, c) = (1 − β, 1 − β + x, 2 − y) followed by the linear transformation in Eq. (C.4). 10.6. Prove Prop. 10.8.9 and Prop. 10.8.8 by repeatedly applying Thm. 10.8.1.  `   10.7. Prove that for s = Cat {2}bj , cj , {2}aj , 1 , ` ≥ 1, we have j=1

 `   ζ ? (s) = ζ ] Cat 2bj + 2, {1}cj −3 , 2aj + 2 . j=1

10.8. Let t, ` ∈ N and aj , bj , cj − 3 ∈ N0 for all j ≥ 1. Prove the following two general formulas.

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 `  Cat {2}bj , cj , {2}aj , 1 , {2}t with t, ` ≥ 1, j=1

 `   ζ ? (s) = −ζ ] Cat 2bj + 2, {1}cj −3 , 2aj + 2 , 2t . j=1

(10.48)

  `  (2-1-2-c-2-1-2): For s = {2}a0 , 1, Cat {2}bj , cj , {2}aj , 1 , {2}t , with j=1

` ≥ 0 and a0 , t ≥ 1, show that  `   ζ ? (s) = −ζ ] 2a0 + 1, Cat 2bj + 2, {1}cj −3 , 2aj + 2 , 2t . j=1

(10.49)

Then verify the following special cases: ζ ? ({2}b , 3, {2}a , 1, {2}t ) = −2ζ(2b + 2a + 2t + 4) − 4ζ(2b + 2a + 4, 2t)

−4ζ(2b + 2, 2a + 2t + 2) − 8ζ(2b + 2, 2a + 2, 2t),

and ζ ? ({2}a1 , 1, {2}b , 3, {2}a2 , 1, {2}t )

= −2ζ(2a1 + 2b + 2a2 + 2t + 5) − 4ζ(2a1 + 2b + 2a2 + 5, 2t)

− 4ζ(2a1 + 2b + 3, 2a2 + 2t + 2) − 8ζ(2a1 + 2b + 3, 2a2 + 2, 2t) − 4ζ(2a1 + 1, 2b + 2a2 + 2t + 4) − 8ζ(2a1 + 1, 2b + 2a2 + 4, 2t)

− 8ζ(2a1 + 1, 2b + 2, 2a2 + 2t + 2) − 16ζ(2a1 + 1, 2b + 2, 2a2 + 2, 2t). 10.9. For any positive integer n, show that ζ ? ({2}n−1 , 3) =

n X l=1

ζ ? ({2}l , 1, {2}n−l ) −

n−1 X l=0

ζ ? ({2}l , 3, {2}n−1−l ).

10.10. For all nonnegative integers m and n, show that ζ ? ({2}m


{3, 1}

n

) ∈ π 2m+4n Q.

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Chapter 11

q -Analogs of Multiple Zeta Functions

In this chapter we define some q-analogs of the multiple zeta functions and study their analytic properties. We will also consider some special values at non-positive integers. The theory of special values at positive integers will be developed in the next chapter. As before, throughout this chapter let q be a real number with 0 < q < 1. 11.1 Definitions First, for positive integer d, we define an auxiliary function of 2d complex variables s1 , . . . , sd , t1 , . . . , td ∈ C: X

ζqt [s] = ζq(t1 ,...,td ) [s1 , . . . , sd ] :=

k1 >···>kd >0

q k1 t1 +···+kd td . [k1 ]s1 · · · [kd ]sd

(11.1)

Again, |s| := s1 + · · · + sd is called the weight and the number d is called the depth, denoted by dp(s). Proposition 11.1.1. The function ζqt [s] converges if Re(t1 + · · · + tj ) > 0 for all j = 1, . . . , d. It can be analytically continued to a meromorphic function over C2d via the series expansion ζqt [s]

= (1 − q)

|s|

∞ X r1 ,...,rd

  d  Y sj + rj − 1 q (d+1−j)(rj +tj ) , (11.2) 1 − q r1 +t1 +···+rj +tj rj =0 j=1

where |s| = s1 + · · · + sd . It has the following (simple) poles: t1 + · · · + tj ∈ 2πi Z≤0 + log q Z for j = 1, . . . , d. Proof. Assume |Re(sj )| < Nj and let τj = Re(tj ) for all j = 1, . . . , d. Since 1 < [k] = 1 + q + · · · + q k−1 < k, 341

(11.3)

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we have t ζq [s]
···>kd >0 j=1 ∞ X

=

d Y

(mj + · · · + md )Nj q (mj +···+md )τj

m1 ,...,md =1 j=1


0 mN q mτ converges if τ > 0. This proves the first part of the proposition. We now prove Eq. (11.2). By the binomial expansion  ∞  X s+r−1 r (1 − x)−s = x r r=0 we get ζqt [s] = (1 − q)|s|

∞ X

X

k1 >···>kd >0 r1 ,...,rd



  d  Y s + r − 1 j j  q kj (rj +tj )  . r j =0 j=1

As 0 < q < 1 the series converges absolutely by Stirling’s formula so we can change the order of the summations. The proposition follows immediately from the next lemma by taking xj = q tj +rj for j = 1, . . . , d. Lemma 11.1.2. Let xj ∈ C such that |xj | < 1 for j = 1, . . . , d. Then X

d Y

k

xj j =

k1 >···>kd >0 j=1

d Y xd+1−j x1 · · · xj j = . 1 − x · · · x 1 − x1 · · · xj 1 j j=1 j=1 d Y

(11.4)

Proof. By re-indexing kj = m1 + · · · + mj we have X

d Y

k

xj j =

∞ X m1 =1

k1 >···>kd >0 j=1

=

···

∞ X md =1

m

d−1 x1m1 +···+md · · · xd−1

+md md xd

d Y

x1 · · · xj . 1 − x1 · · · xj j=1

The second equation in the lemma follows immediately. Now we define the first version of the q-analog of the multiple zeta functions.

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Definition 11.1.3. For all s = (s1 , . . . , sd ) ∈ Cd , we set

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ζqI [s] := ζqs−1 [s] =

X k1 >···>kd ≥1

d Y q kj (sj −1) . [kj ]sj j=1

(11.5)

The following corollary follows easily from Prop. 11.1.1. Theorem 11.1.4. Set   2πi 2πi    s1 ∈ 1 +  Z, or s ∈ Z + Z , 1 ≤0 = 6 0   log q log q 0 d Sd := (s1 , . . . , sd ) ∈ C .  or s1 + · · · + sj ∈ Z≤j + 2πi Z, j > 1      log q Then ζqI [s1 , . . . , sd ] defined by Eq. (11.5) can be extended to a meromorphic function with simple poles lying along S0d and it can be written as ζqI [s]

= (1 − q)

|s|

+∞ X r1 ,...,rd

  d  Y sj + rj − 1 q (d+1−j)(rj +sj −1) . 1 − q r1 +s1 +···+rj +sj −j rj =0 j=1

(11.6)

It is easy to see that as q → 1 the set S0d becomes the set defined by Eq. (1.15): ( ) s =1, s + s = 2, 1, −2m ∀m ∈ Z , 1 2 ≥0 d 1 Sd = (s1 , . . . , sd ) ∈ C (11.7) and s1 + · · · + sj ∈ Z≤j ∀j ≥ 3. which is exactly the singular set of the multiple zeta function ζ(s1 , . . . , sd ). To see the effect of taking different specializations of tj in ζqt [s] we define the shifting operators Sj (1 ≤ j ≤ d) on any function fq [s1 , . . . , sd ] as follows: Sj fq [s1 , . . . , sd ] := fq [s1 , . . . , sd ] + (1 − q)fq [s1 , . . . , sj − 1, . . . , sd ]. (11.8) It is obvious that these operators commute. Proposition 11.1.5. Let n1 , . . . , nd ∈ N0 . Then we have ζq(s1 −1−n1 ,...,sd −1−nd ) [s1 , . . . , sd ] = S1n1 ◦ · · · ◦ Sdnd ζqI [s1 , . . . , sd ]   nd n1 d   X X Y n j  = ··· (1 − q)rj  ζqI [s1 − r1 , . . . , sd − rd ]. r j r =0 r =0 j=1 1

d

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Proof. We only sketch the proof in the case n2 = · · · = nd = 0. The general case is completely similar. Suppose n1 = n = 1. Then ζq(s1 −2,s2 −1,...,sd −1) [s1 , . . . , sd ] X q k1 (s1 −2)+k2 (s2 −1)+···+kd (sd −1) = [k1 ]s1 · · · [kd ]sd k1 >···>kd >0

=

X k1 >···>kd >0

q k1 (s1 −2) (1 − q k1 ) + q k1 (s1 −1) q k2 (s2 −1)+···+kd (sd −1) · [k1 ]s1 [k2 ]s2 · · · [kd ]sd

= S1 ζqI [s1 , . . . , sd ]. The rest follows easily by induction on n and some binomial identities. See Exercise 11.1. Corollary 11.1.6. Let n be a positive integer. Then n   X n (s−1−n) n I (1 − q)r ζqI [s − r]. ζq [s] = S ζq [s] = r r=0 Observe that one effect of the shifting operator is to bring in more poles. Essentially, Sn shifts all the poles of ζqI [s] by n to the right on the complex plane. 11.2 Analytic Continuation As a golden rule, the correct q-analog of any classical mathematical object should have the property that when q → 1 the q-analog becomes its classical counterpart. Although it is obvious that when Re(s1 + · · · + sj ) > j we have lim ζqI [s1 , . . . , sd ] = ζ(s1 , . . . , sd ),

q→1

it is highly nontrivial to verify this still holds for all (s1 , . . . , sd ) outside the singular set Eq. (1.15) of the analytic continuation of the multiple zeta function ζ(s1 , . . . , sd ). We will prove this fact in this section. Similar to the classical case (see Sec. 1.3) the basic idea is to use the classical Euler–Maclaurin summation formula Eq. (1.12). Let f (x) be any (complex-valued) C ∞ function on [1, ∞) and let m and M be two positive integers. Recall that Z m m M  X X 1 Br+1  (r) f (n) = f (x) dx+ (f (1)+f (m))+ f (m)−f (r) (1) 2 (r + 1)! 1 n=1 r=1 Z (−1)M +1 m e − BM +1 (x)f (M +1) (x) dx (11.9) (M + 1)! 1

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where the periodic Bernoulli polynomial satisfies

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ek (x) = −k! B

X n∈Z\{0}

e2πinx . (2πin)k

(11.10)

We first consider the depth one case. Theorem 11.2.1. The q-analog of the Riemann zeta function ζqI [s] can be extended to a meromorphic function on C \ S01 with simple poles along S01 . Further, for all s ∈ C \ S1 lim ζqI [s] = ζ(s).

q→1

Proof. We set s − 1 + qx q x(s−1) , F 0 (x) = (log q)q x(s−1) , x s (1 − q ) (1 − q x )s+1 s(s + 1) − 3s(1 − q x ) + (1 − q x )2 F 00 (x) =(log q)2 . q x(1−s) (1 − q x )s+2 F (x) =

Taking f (x) = F (x) in Eq. (11.9) and letting m → ∞, we get Z ∞ x(s−1) ∞ X q n(s−1) q s−1 1 (log q)q s−1 s − 1 + q q · = dx + − (1 − q n )s (1 − q x )s 2 (1 − q)s 12 (1 − q)s+1 1 n=1 Z s(s + 1) − 3s(1 − q x ) + (1 − q x )2 (log q)2 ∞ e B2 (x) dx − 2 q x(1−s) (1 − q x )s+2 1 for Re(s) > 1. Hence (see Exercise 11.2) ζqI [s] =

q s−1 q − 1 q s−1 q s−1 log q · + + (s − 1 + q) s − 1 log q 2 12 q − 1 Z (log q)2 ∞ e s(s + 1) − 3s(1 − q x ) + (1 − q x )2 − (1 − q)s dx. B2 (x) 2 q x(1−s) (1 − q x )s+2 1 (11.11)

By Eq. (11.10) q s−1 q − 1 q s−1 q s−1 log q · + + (s − 1 + q) s − 1 log q 2 12 q − 1 x x 2 X (1 − q)s (log q)2 Z ∞ 2πinx s(s + 1) − 3s(1 − q ) + (1 − q ) e dx. (2πin)2 q x(1−s) (1 − q x )s+2 1

ζqI [s] = +

n∈Z\{0}

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Setting q x = u we find

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ζqI [s] =

q s−1 q − 1 q s−1 q s−1 log q · + + (s − 1 + q) s − 1 log q 2 12 q − 1 X (1 − q)s log q X aν (s)bq (s − 1 + δn, −s + ν), (11.12) − (2πin)2 ν=±1,0 n∈Z\{0}

where δ = 2πi/ log q, a−1 (s) = s(s + 1), a0 (s) = −3s, a1 (s) = 1, and Z t uα−1 (1 − u)β−1 du. (11.13) bt (α, β) = 0

Observe that σ = Re(s) > 1 and for all ν = −1, 0, 1 Z q uσ−2 (1 − u)−σ+ν−1 du. bq (s − 1 + δn, −s + ν) ≤ 0

Hence the sum in Eq. (11.12) converges absolutely. For Eq. (11.13), we have (see Exercise 11.3) bq (α, β) =

M −1 X

(−1)r−1

r=1

(1 − β)r−1 α+r−1 q (1 − q)β−r (α)r

(1 − β)M −1 bq (α + M − 1, β − M + 1) (α)M −1

+ (−1)M −1

(11.14)

for any M ≥ 2. Applying this to bq (s − 1 + δn, −s − 1) and using q δn = 1 we get bq (s − 1 + δn, −s + ν) = + (−1)M −1

M −1 X

(−1)r−1

r=1

(s + 1 − ν)r−1 s+r−2 q (1 − q)−s+ν−r (s − 1 + δn)r

(s + 1 − ν)M −1 bq (s − 2 + M + δn, −s + ν + 1 − M ). (s − 1 + δn)M −1 (11.15)

For all ν = −1, 0, 1, this provides the analytic continuation of bq (s − 1 + δn, −s + ν) as functions of s into the region Re(s) > 2 − M . This implies that X s(s + 1) bq (s − 1 + δn, −s − 1) (2πin)2 n∈Z\{0}

=

M −1 X

X

r=1

n∈Z\{0}

(−1)r−1

(s)r+1 · q s+r−2 (1 − q)−s−1−r (2πin)2 (s − 1 + δn)r

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− (−1)

X

M −1

n∈Z\{0}

Observe that log q lim = −1, q→1 1 − q

(s)M +1 · log q (2πin)2 (s − 1 + δn)M −1

347

∞

Z

mzv-mpl

1

e2πinx q x(s−2+M ) dx. (1 − q x )s+M +1

lim (1 − q)r (s − 1 + δn)r = (−2πi)r ,

q→1

1 − qx = x. q→1 1 − q lim

Thus for Re(s) > 2 − M , by Eq. (11.10) (and its special case when x = 1), we see that X s(s + 1) bq (s − 1 + δn, −s − 1) lim (1 − q)s log q q→1 (2πin)2 n∈Z\{0}

=

M −1 X

X

r=1 n∈Z\{0}

=−

M −1 X r=1

(s)r+1 − (2πin)r+2

X n∈Z\{0}

Br+2 (s)M +1 (s)r+1 + (r + 2)! (M + 1)!

(s)M +1 (2πin)M +1 Z

Z

∞

e2πinx x−s−M −1 dx

1

∞

eM +1 (x)x−s−M −1 dx. B

1

The same procedure applies to the sums involving bq (s − 1 + δn, −s) and bq (s − 1 + δn, −s + 1). See Exercise 11.4. Consequently, for Re(s) > 2 − M , we obtain M

lim ζqI [s] =

q→1

1 X Br+1 1 + + (s)r s − 1 2 r=1 (r + 1)! Z ∞ (s)M +1 eM +1 (x)x−s−M −1 dx = ζ(s) B − (M + 1)! 1

by Eq. (1.16). This proves the theorem since M is arbitrary . Theorem 11.2.2. The multiple q-zeta function ζqI [s1 , . . . , sd ] can be extended to a meromorphic function on Cd \ S0d with simple poles along S0d . Further, for all (s1 , . . . , sd ) ∈ Cd \ Sd lim ζqI [s1 , . . . , sd ] = ζ(s1 , . . . , sd ).

q→1

Proof. We proceed by induction on d. If d = 1 the theorem is reduced to Thm. 11.2.1. Now we assume d ≥ 2. In definition Eq. (11.5) we replace s1 , k1 , and k2 by s, n and k, respectively. Taking f (x) = F (x + k − 1) and letting m → ∞ in Eq. (1.12) we get ! ∞ X q n(s−1) X s = (1 − q) −F (k) + f (n) [n]s n=1 n>k

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= (1 − q)s

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∞

Z k

1 1 1 F (x) dx − F (k) − F 0 (k) − 2 12 2

Z

∞

e2 (x)f 00 (x) dx B



1

1 q k(s−1) 1 log q q k(s−1) (s + q k − 1) (q − 1) q k(s−1) − + = (s − 1) log q [k]s−1 2 [k]s 12 q − 1 [k]s+1 s 2 Z ∞ x x 2 (1 − q) (log q) e2 (x)q x(s−1) s(s + 1) − 3s(1 − q ) + (1 − q ) dx − B 2 (1 − q x )s+2 k (11.16) e2 (x+k−1) = B e2 (x) by periodicity. By exactly the same argument because B as in the proof of depth 1 case, we can obtain from Eq. (11.10) the following expression for the last term involving the integral in Eq. (11.16): X (1 − q)s log q X (11.17) aν (s)bqk (s − 1 + δn, −s + ν). − (2πin)2 ν=±1,0 n∈Z\{0}

For simplicity, write ζ[s] = ζqI [s] and S = S1 . Set s0 = (s3 , . . . , sd ) if d ≥ 3 and s0 = ∅ if d = 2. Replacing q by q k in Eq. (11.15), plugging it into Eq. (11.17) which is used to replace the last line of Eq. (11.16), and then applying Prop. 11.1.5, we finally get (q − 1) 1 ζ[s1 + s2 − 1, s0 ] − Sζ[s1 + s2 , s0 ] (s1 − 1) log q 2 X s1 log q 2 log q + S ζ[s1 + s2 + 1, s0 ] + Sζ[s1 + s2 , s0 ] − (Cν + Dν ), 12 q − 1 12 ν=±1,0

ζ[s1 , s2 , s0 ] =

(11.18) where Cν and Dν are the contributions from the sum involving bqk (. . . , −s+ ν) (note that s = s1 ). Explicitly they are computed as follows. Write T (q, s, n, r) =

r−1 Y

2πin + (s − 1 + j) log q

j=0


−1

.

(11.19)

Set M 0 = M − 1. Then 0

C−1 =

M X

X

r=1 n∈Z\{0}

D−1 = −

X k2 >···>kd >0

 ×

T (q, s, n, r) (2πin)2

log q q−1



log q q−1

r+1

q k2 (s2 −1)+···+kd (sd −1) [k2 ]s2 · · · [kd ]sd

M +1 (s)M +1 ·

Z

(s)r+1 · S3 ζ[s1 + s2 + r + 1, s0 ], X

n∈Z\{0}

T (q, s, n, M 0 ) × (2πin)2

∞

k2

e2πinx q x(s−2+M )



1 − qx 1−q

−s−M −1 dx

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q k3 (s3 −1)+···+kd (sd −1) [k3 ]s3 · · · [kd ]sd

X

=:

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k3 >···>kd >0

∞ X

349

R(M, q, k, s2 , s),

k=k3 +1

where we replace the index k2 by k. Similarly, M0 X X T (q, s, n, r)  log q r (s)r · S2 ζ[s1 + s2 + r, s0 ], C0 = 3 log q 2 (2πin) q − 1 r=1 n∈Z\{0}

D0 = − 3 log q  ×

X k2 >···>kd >0

log q q−1

M (s)M ·

q k2 (s2 −1)+···+kd (sd −1) [k2 ]s2 · · · [kd ]sd Z

∞

e

2πinx x(s−2+M )

X n∈Z\{0}



q

k2

T (q, s, n, M 0 ) (2πin)2

1 − qx 1−q

−s−M dx,

and 0

2

C1 = (log q)

M X

X

r=1 n∈Z\{0}

T (q, s, n, r) (2πin)2



log q q−1

r−1

× (s)r−1 · Sζ[s1 + s2 + r − 1, s0 ], X q k2 (s2 −1)+···+kd (sd −1) D1 = − (log q)2 [k2 ]s2 · · · [kd ]sd k2 >···>kd >0

M 0

X n∈Z\{0}

T (q, s, n, M 0 ) (2πin)2

−s−M +1 1 − qx (s)M 0 · dx. e q × 1−q k2 P∞ The crucial step next is to control the summations k=k3 +1 and show that they converge uniformly with respect to q. When 0 < q ≤ 1/2 this is clear. The only non-trivial part is when q → 1. So we assume 1/2 < q < 1. Note that log q 1 lim T (q, s, n, r) = , lim = 1. q→1 q→1 q − 1 (2πin)r 

log q q−1

Z

∞

2πinx x(s−2+M )



Lemma 11.2.3. Let s1 = s = σ + iτ . Let q0 = max{1/2, e(6−2π)/τ } if τ > 0 and let q0 = 1/2 if τ ≤ 0. Then for all 1 > q > q0 and positive integer k we have log q 1 and |T (q, s, n, r)| < . q − 1 < 2, (6n)r Proof. Let f (q) = 2(1 − q) + log q. Then f 0 (q) = −2 + 1/q < 0 whenever q > q0 . So f (q) > f (1) = 0 whenever 1 > q > q0 . This implies 2(1 − q) > − log q whence log q/(q − 1) < 2.

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To bound T (q, s, n, r) we consider each of its factors in definition Eq. (11.19). For each 0 ≤ j < r, we have
2 |2πin+(s−1+j) log q|2 = (σ−1+j) log q +(2πn+τ log q)2 ≥ (2πn+τ log q)2 which is independent j. If τ ≤ 0 then clearly |2πin + (s − 1 + j) log q| > 6n. If τ > 0 then it follows from q > e(6−2π)τ that 2πn + τ log q > 2πn + 6 − 2π ≥ 6n, as desired. Next we want to bound the integral terms in D−1 . Let |σ1 | < N and |σ2 | < N 0 for some positive integers N and N 0 . Fix an arbitrary x > k and a positive integer M > 16 + 2N + 6

d X

(Nj + 1).

(11.20)

j=2

Then q

−s−M −1  k(s0 −1) x(s−2+M ) 1 − q x q q 1−q  M −N +1 1−q < q x(M −N −2)−kM/6 . 1 − qx

−k(M/6−N 0 −1)

Denote by g(q) the right-hand side of the above inequality. It can be proved that for 1/2 < q < 1, g(q) is increasing as a function of q so that (see Exercise 11.5) 1 (11.21) g(q) ≤ lim g(q) = M −N +1 . q→1 x We now can bound the innermost sum of D−1 . From Lemma 11.2.3 and Eq. (11.21) we have (if d = 2 then take k3 = 0) ∞ X k=1+k3

|R(M, q, k, s0 , s)| ∞ X

0 2ζ(M + 1) M +1 2 (N )M +1 q k(M/6−N −1) 2 M −1 4π 6 k=1+k3   X 0 ∞ (M + 1)! M + N q k(M/6−N −1) < M −N M +1 k M −N −N 0


2. Therefore by Eq. (11.3)   d X Y N D−1 < (2M )2M  kj j q kj (−Nj −1)  q k3 (M/6−N2 −1) (11.22) j=3

k3 >···>kd

which converges by the condition Eq. (11.20) on M (see the proof of Prop. 11.1.1). Exactly the same argument applies to the integral terms in D0 and D1 (see Exercise 11.7). These convergence results imply two things. First we can show by induction on d that Eq. (11.18) gives rise to an analytic continuation of ζ[s1 , . . . , sd ] as a meromorphic function on Cd \ S0d . Second, also by induction on d, we now can conclude that it’s legitimate to take the limit q → 1 inside the sums of Cν and Dν to get (note that limq→1 Sn ζ[s] = ζ(s) for any s ∈ Cd−1 \ S0d−1 and any positive integer n) lim ζ[s1 , s2 , s0 ]

q→1

=

1 1 s ζ(s1 + s2 − 1, s0 ) − ζ(s1 + s2 , s0 ] + ζ(s1 + s2 + 1, s0 ) s−1 2 12 M −1 X X 1 − (s ) · ζ(s1 + s2 + r + 1, s0 ) r+2 1 r+1 (2πin) r=1 n∈Z\{0} Z ∞ X X 1 (s)M +1 e2πinx x−s1 −M −1 dx + k2s2 · · · kdsd (2πin)M +1 k2 k2 >···>kd

=

n∈Z\{0}

M +1 X

Br (s1 )r−1 · ζ(s1 + s2 + r − 1, s3 , . . . , sd ) r! r=0 Z ∞ X 1 1 eM +1 (x) (s)M +1 dx − B (M + 1)! k2s2 · · · kdsd k2 xs1 +M +1 k2 >···>kd >0

by Eq. (11.10) and its specialization with x = 1 X er+2 (1) B Br+2 1 =− =− . r+2 (2πin) (r + 2)! (r + 2)! n∈Z\{0}

The theorem now mostly follows from Thm. 1.3.2 (by replacing M by M +1 2πi there). The poles at s1 = m − log q n are given by the first term in formula Eq. (11.18) when m = 1 and n = 0 and by the terms T (q, sd , n, r) as defined in Eq. (11.19) if m ≤ 1 and n 6= 0. The location of the other poles are obtained by induction using those poles of the q-Riemann zeta function presented in Thm. 11.2.1 for the initial step. This completes the proof of the theorem.

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11.3 Special Values at Non-Positive Integers

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In this section, we will consider some special values of ζqI [s1 , . . . , sd ] at nonpositive integers. Theorem 11.3.1. For any nonnegative integer m, the limiting value ζqI [−m] = lims→−m ζqI [s] exists and is given explicitly by ) (m   m+1 X m 1 (−1) ζqI [−m] = (1 − q)−m (−1)r + . r q m+1−r − 1 (m + 1) log q r=0 Moreover, for all m ∈ N, we have lim ζqI [−m] = −

q→1

Bm+1 . m+1

(11.23)

Proof. Note the terms with the sum Eq. (11.6) vanishes when
r ≥ m + 2 in−m+r−1 s → −m since −m+r−1 = 0 and 1 − q 6= 0. On the other hand, r for r = m + 1 we have lims→−m (s + m)/(1 − q s+m ) = −1/ log q and hence    s+m  (−1)m m! 1 (−1)m+1 q s+m = − = . lim s→−m m + 1 1 − q s+m (m + 1)! log q (m + 1) log q To prove Eq. (11.23) we apply Thm. 11.2.1 to get lim ζqI [−m] = ζ(−m) = −

q→1

Bm+1 . m+1

This completes the proof of the theorem. Remark 11.3.2. Thanks to the theorem, we can regard mζqI [1 − m] as the q-analog of the Bernoulli number Bm . In general, for integers n1 , . . . , nd we can define ζqI [n1 , . . . , nd ] = lim · · · lim ζqI [s1 , . . . , sd ], sd →nd

ζqR [n1 , . . . , nd ]

s1 →n1

= lim · · · lim ζqI [s1 , . . . , sd ] s1 →n1

sd →nd

if the limits exist. Similarly we can define ζ(n1 , . . . , nd ) = lim · · · lim ζ(s1 , . . . , sd ), sd →nd

s1 →n1

R

ζ (n1 , . . . , nd ) = lim · · · lim ζ(s1 , . . . , sd ) s1 →n1

sd →nd

if the limits exist. Interesting phenomena occur already in the case d = 2 and these can be generalized to arbitrary depth. By Thm. 11.1.4 we get

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ζqI [s1 , s2 ]   +∞  X s1 + r1 − 1 s2 + r2 − 1 (1 − q)s1 +s2 q 2s1 +2r1 +s2 +r2 −3 (1 − q s1 +r1 −1 )(1 − q s1 +r1 +s2 +r2 −2 ) r1 r2 r1 ,r2 =0  q 2s1 +s2 −3 s2 q 2s1 +s2 −2 s1 +s2 = (1 − q) + (1 − q s1 −1 )(1 − q s1 +s2 −2 ) (1 − q s1 −1 )(1 − q s1 +s2 −1 ) 2s1 +s2 −1 s1 q s1 s2 q 2s1 +s2 + + (1 − q s1 )(1 − q s1 +s2 −1 ) (1 − q s1 )(1 − q s1 +s2 )  s2 (s2 + 1)q 2s1 +s2 −1 s1 (s1 + 1)q 2s1 +s2 +1 + + + · · · . (11.24) 2(1 − q s1 −1 )(1 − q s1 +s2 ) 2(1 − q s1 +1 )(1 − q s1 +s2 ) It is straightforward to get 1 1 3 − ζqI [0, 0] = 2 + , (q − 1)(q − 1) 2(q − 1) log q log2 q 1 1 q ζqR [0, 0] = 2 − + . (q − 1)(q − 1) (q − 1) log q 2(q − 1) log q Taking limit we can find 1 5 lim ζqI [0, 0] = , lim ζqR (0, 0) = . q→1 q→1 3 12 This is consistent with the behavior of the double zeta functions near (0, 0) (see Exercise 1.6). It has the following asymptotic expansion: 5s1 + 4s2 ζ(s1 , s2 ) = + R(s1 , s2 ) 12(s1 + s2 )

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=

where R(s1 , s2 ) is analytic at (0, 0) and lim(s1 ,s2 )→(0,0) R(s1 , s2 ) = 0. Let n, k be two nonnegative integers, and m = k − n − 2. We now consider the double zeta function around (s1 , s2 ) = (−n, −m) which has the following expression by Thm. 1.3.2(ii): n+1 X Br  n  −1 1 ζ(s1 + s2 − 1) − ζ(s1 + s2 ) − ζ(s1 + s2 + r − 1) n+1 2 r r−1 r=2 Bk n!(k − n − 2)!(s1 + n)ζ(s1 + s2 + k − 1) (11.25) k! where α(m) = 0 if m ≤ −1 and α(m) = 1 if m ≥ 0. Note that the last term is zero when computing + α(m)(−1)n

ζ(−n, −m) =

lim

lim ζ(s1 , s2 )

s2 →−m s1 →−n

while it has possibly nontrivial contribution for ζ R (−n, −m), since lim

lim (s1 + n)ζ(s1 + s2 + k − 1) = 1.

s1 →−n s2 →−m

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Table 11.1: Poles and indeterminacy of the double zeta function, ak =
n Bk−1 /(2k − 2) and bn,k = k−1 Bk /k. k, n

pole, residue

k=0 k=1

−1/(n + 1) −1/2

2 - k, 3 ≤ k ≤ n + 1

NO

2 - k, k = n + 2 2 - k, k > n + 2 2|k, 2 ≤ k ≤ n + 1

NO NO (−1)k+1 bn,k

indeterminacy ζ=ζ R (−n,n+2−k)

NO NO ak +

n+1 X r=k−1

bn,r ζ(r + 1 − k) 2ak ak NO

When m ≥ 0 and k is even the values of ζ(−n, −m) and ζ R (−n, −m) are usually different: n+1 X Br  n  Bk−r Bk ζ(−n, −m) = + , (11.26) k(n + 1) r=1 r r − 1 k − r

Bk n!(k − n − 2)!. (11.27) k! Note that the term corresponding to r = 1 is non-zero if and only if k = 2 (and n = 0). From this observation we again recover that 5 1 . (11.28) ζ(0, 0) = , ζ R (0, 0) = 3 12 We now consider the double q-zeta function. ζ R (−n, −m) =ζ(−n, −m) + (−1)n

Theorem 11.3.3. Let k, n be two nonnegative integers, and m = k −n−2. If m ≤ −1 then the double q-zeta function ζqI [s1 , s2 ] has a pole at (−n, −m) with residue given by: Res

(s1 ,s2 )=(−n,−m)

ζqI [s1 , s2 ]

−(1 − q)2−k (log q)−1     k  n X r n+1−r   (−1) /(q n+1−r − 1)  k − r r = r=0   n X  (−1)n  r n n+1−r  − 1) −   (−1) r /(q (n + 1) log q r=0 Proof. This follows directly from Eq. (11.24).

if k ≤ n, if k = n + 1.

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Corollary 11.3.4. Let n be a nonnegative integer. Then Res

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(s1 ,s2 )=(−n,1)

ζqI [s1 , s2 ] =

q−1 I ζ [−n]. log q q

(11.29)

Setting B1∗ = 1/2 and Bn∗ = Bn for all n > 1, then for all n ≥ 0 we have lim ζqI [−n] = −

q→1

∗ Bn+1 = Res ζ(s1 , s2 ). n + 1 (s1 ,s2 )=(−n,1)

(11.30)

Proof. Equation Eq. (11.29) follows from the case m = −1 in the above theorem and Thm. 11.3.1. The first equality in Eq. (11.30) is Eq. (11.23) and the second equality follows from Table 11.1. Corollary 11.3.5. Let k, n be two nonnegative integers, m = k − n − 2 ≤ −2. Then lim

ζqI [s1 , s2 ] =

Res

q→1 (s1 ,s2 )=(−n,−m)

Res

ζ(s1 , s2 ).

(s1 ,s2 )=(−n,−m)

(11.31)

Proof. By Thm. 11.3.3 and Table 11.1 we only need to prove lim

q→1

   k X (q − 1)1−k (−1)r n + 1 − r n r=0

q n+1−r − 1

  (−1)k   n +1  = B n k    k k−1

k−r

r

(11.32)

if k = 0, (11.33) if 1 ≤ k ≤ n.

First, by the generating function of the Bernoulli numbers 1 q n+1−r

−1

=

1 e(n+1−r) log q

−1

=

∞ X Bl l=0

l!

(n + 1 − r) log q


l−1

.

Plugging this into the left-hand side of Eq. (11.32), replacing 1 − q by − log q, and changing the order of the summations we get    k X n r n+1−r (log q) (−1) /(q n+1−r − 1) (11.34) k − r r r=0    k ∞ X X n Bl l−k r n+1−r = (log q) (−1) (n + 1 − r)l−1 . (11.35) l! k − r r r=0 1−k

l=0

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Then the inner sum over r is the coefficient of xk of the following polynomial    n+1 k XX n r n+1−r fl (x) = (−1) (n + 1 − r)l−1 xk k − r r k=0 r=0   n n+1 X X n + 1 − r n = (−1)r (n + 1 − r)l−1 xk r k − r r=0 k=r   n X n = (x + 1)n+1 (−y)r (n + 1 − r)l−1 , (11.36) r r=0 where y = x/(x + 1). When l = 0 this expression becomes   n n+1 (x + 1)n+1 X (−y)r f0 (x) = r n + 1 r=0 h i i (x + 1)n+1 1 h = (1 − y)n+1 − (−y)n+1 = 1 − (−x)n+1 . n+1 n+1 Note that k ≤ n we see the coefficient of xk in f0 (x) is 0 if k > 0 and it’s 1/(n + 1) if k = 0. If k = 0 then only the constant term −1/(n + 1) in Eq. (11.34) remains when q → 1 which proves the corollary in this case. So we can assume l, k > 0. Then l−1 n    n o X d n+1 r n n+1−r (x + 1) (−y) z fl (x) = z dz r r=0 z=1 l−1 n  o d (x + 1)n+1 z(z − y)n = z . dz z=1 Note that highest degree term in fl (x) is contained in  l−1 d n n+1 (z − y) (x + 1) dz z=1

= n(n − 1) · · · (n − l + 2)(x + 1)n+1 (1 − y)n−l+1 = n(n − 1) · · · (n − l + 2)(x + 1)l .

If l = 1 one can easily modify this to get just x + 1. If l < k then the coefficient of xk in fl (x) is 0. If l = k it is equal to   n n(n − 1) · · · (n − k + 2) = (k − 1)! . k−1 The last express is valid even for k = l = 1. Thus the range of l in the outer sum of Eq. (11.34) starts from k. Moreover, the first term of Eq. (11.34) is   Bk n k k−1 as desired. This completes the proof of the corollary.

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Proposition 11.3.6. Let k, n be two nonnegative integers such that n ≥ k and k is even. Let m = k − n − 2. Then Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 06/12/16. For personal use only.

−f (q)(q − 1)/ log q , D(q)

ζqI [s1 , s2 ] =

Res

(s1 ,s2 )=(−n,−m)

n+1 Y

D(q) =

F (q, j)j ,

j=n+1−k


where f (q) ∈ Z[q] is palindromic with leading coefficient nk , F (q, j) ∈ Z[q] is a factor of (q j − 1)/(q − 1), j = 0 or 1, and degq D(q) = n + degq f (q), such that lim

Res

q→1 (s1 ,s2 )=(−n,−m)

ζqI [s1 , s2 ] =

Res

(s1 ,s2 )=(−n,−m)

ζ(s1 , s2 ).

Proof. The computational proof is left as an exercise for the interested reader. See Exercise 11.6. Example 11.3.7. By Thm. 11.3.3 we find with the help of Maple ζqI [s1 , s2 ] =

Res

(s1 ,s2 )=(−4,4)

where Pa (q, m) = lim

Pm

j=0

Res

−2q 3 (3q 2 + 4q + 3)(q − 1)/ log q P1 (q, 2)P1 (q, 3)P1 (q, 4)

q aj . Moreover we can check that ζqI [s1 , s2 ] =

q→1 (s1 ,s2 )=(−4,4)

Res

(s1 ,s2 )=(−4,4)

ζ(s1 , s2 ) = −

1 3

by Table 11.1 with k = 2 and n = 4. Example 11.3.8. By Thm. 11.3.3, we obtain Res

(s1 ,s2 )=(−8,6)

ζqI [s1 , s2 ] =

−14q 5 g(q)(q − 1)/ log q P1 (q, 4)P1 (q, 5)P1 (q, 6)P2 (q, 3)P3 (q, 2)

where g(q) is a polynomial in q of degree 14 satisfying q 14 g(1/q) = g(q) = 5q 14 +6q 13 +8q 12 +7q 11 −q 10 −20q 9 −30q 8 −34q 7 −· · · . Then we can compute with Maple lim

Res

q→1 (s1 ,s2 )=(−8,6)

ζqI [s1 , s2 ] =

Res

(s1 ,s2 )=(−8,6)

ζ(s1 , s2 ) =

7 15

by Table 11.1 with k = 4 and n = 8. Example 11.3.9. Consider the point (s1 , s2 ) = (−9, 5). We have Res

ζqI [s1 , s2 ] =

(s1 ,s2 )=(−9,5)

−42q 4 g(q)(q − 1)/ log q P1 (q, 4)P1 (q, 6)P1 (q, 7)P2 (q, 2)P3 (q, 2)A(q, 4)

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where A(q, m) = satisfying

Pm

j=0 (−1)

j j

q and g(q) is a polynomial in q of degree 18

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q 18 g(1/q) = g(q) = 2q 18 − q 17 − 7q 15 − 11q 14 − 16q 13 − 4q 12 + 9q 11 + 28q 10 + 30q 9 + · · · so that we again have the equality lim

ζqI [s1 , s2 ] =

Res

q→1 (s1 ,s2 )=(−9,5)

Res

(s1 ,s2 )=(−9,5)

ζ(s1 , s2 ) = −

1 2

by Table 11.1 with k = 6 and n = 9. Theorem 11.3.10. Let m, n be two nonnegative integers and k = m+n+2. Then ζqI [s1 , s2 ] has indeterminacy at (−n, −m) such that ζqI [−n, −m](1 − q)k−2    m X (−1)k (−1)r+n+1 m 1 = + 2 m+1−r (m + 1)(n + 1)(log q) (n + 1) log q r q −1 r=0   n X (−1)r+m+1 m!(n + 1 − r)! n 1 + n+1−r − 1 log q (k − r)! r q r=0 )    n m X X n 1 1 r1 +r2 m (−1) + , r1 r2 q n+1−r2 − 1 q k−r1 −r2 − 1 r =0 r =0 1

2

and   m X 1 (−1)r+n+1 m m+1−r − 1 r (n + 1) log q q r=0   m r+n X (−1) k − n − 2 n!(m + 1 − r)! q m+1−r + log q r (k − r)! q m+1−r − 1 r=0 )    m X n X n 1 1 r1 +r2 m (−1) + . r1 r2 q n+1−r2 −1 q k−r1 −r2 −1 r =0 r =0 (

ζqR [−n, −m]

2−k

=(1 − q)

1

2

Proof. This follows from Eq. (11.24) by straightforward computation. Similar to Cor. 11.3.4 and Cor. 11.3.5 we have Corollary 11.3.11. Let m and n be two nonnegative integers. Then lim ζqI [−n, −m] = ζ(−n, −m),

q→1

lim ζqR [−n, −m] = ζ R (−n, −m).

q→1

(11.37)

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359

Proof. Set k = m+n+2. We consider ζqR [−n, −m] first. From Thm. 11.3.10 we get 2  q−1 R −k (A + B + C), ζq [−n, −m] = (q − 1) log q where ∞ X Bj

  m X (−1)m+r+1 m

(m + 1 − r)j−1 , r r=0 j=0   ∞ m X X Bi i r+m+i m n!(m + 1 − r)! B := (log q) (−1) (m + 1 − r)i−1 , i! (k − r)! r r=0 i=0    m X n ∞ X X n Bi Bj i+j k+r1 +r2 m (log q) (−1) C= i!j! r1 r2 r =0 r =0 i,j=0 A :=

j!

(log q)

j

n+1

1

× (n + 1 − r2 )

i−1

2

(k − r1 − r2 )j−1 .

We first compute B as follows. Write B=

∞ X Bi i=0

i!

(log q)i Wim,n ,

where Wim,n :=

m X (−1)r+k+n+i r=0

m!n! (m + 1 − r)i . r!(k − r)!

If i = 0 then we can prove by decreasing induction on n that W0m,n =

m X

(−1)r+k+n

r=0

1 m!n! = . r!(k − r)! k(n + 1)

(11.38)

This is trivial if n = k − 2. Suppose Eq. (11.38) is true for n ≥ 1 then W0m,n−1 = −

k−n−1 (n − 1)! 1 W0 + = , n (n + 1)! kn

as desired. In the rest of the proof, we write Wi = Wim,n . Similarly, we can compute C as follows. Put C=

m ∞ n X Bi Bj (log q)i+j X X (−1)k+r1 +r2 i!j!(n + 1) r1 =0 r2 =0 i,j=0    m n+1 × (n + 1 − r2 )i (k − r1 − r2 )j−1 . r1 r2

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We now change the upper limit of r2 from n to n + 1 in the above. The extra terms correspond to those by setting i = 0 and r2 = n + 1, which produce exactly A. Therefore, ∞ X Bi Bj (log q)i+j Vi,j − A C= i!j! i,j=0 where Vi,j :=

m n+1 X X (−1)k+r1 +r2 mn + 1 (n + 1 − r2 )i (k − r1 − r2 )j−1 . n + 1 r r 1 2 r =0 r =0 1

2

For j ≥ 1, we have  j−1 (  i n X m n+1 X (−1)k+r1 +r2 d d Vi,j = x y dx dy n+1 r1 =0 r2 =0 )    m n + 1 n+1−r2 k−r1 −r2 o × y x r1 r2 y=1 x=1 )  j−1 (  i n o k d d (−1) n+1 k−n−2 x y x(xy − 1) (x − 1) = n+1 dx dy y=1

x=1

(11.39)   0,

if i + j < k; (−1)k n!  (k − i − 1)!, if i + j = k, i ≤ n + 1.  (n + 1 − i)! We have used the fact that if i + j = k and i > n + 1 then j < k − n − 1 and by exchanging the two operators x(d/dx) and y(d/dy) we can easily show that Eq. (11.39) is zero. So if l < k the total contribution to the coefficient of (log q)l from Vi,j with j > 0 is trivial and if l = k it is equal to =

n+1 X

Bi Bk−i

i=0

=

(−1)k n!(k − i − 1)! i!(k − i)!(n + 1 − i)!

 Bk−1     2(k − 1)

Bk     k(n + 1) +

if k is odd, n+1 X i=1



(11.40)



Bi Bk−i n i(k − i) i − 1

if k is even,

because k ≥ n + 2 ≥ 2 and Bk = 0 if k is odd. To deal with Vi,0 , note that Eq. (11.39) still makes sense if we interpret the operator x(d/dx)−1 as follows:  −1 n Z 1 o d F (x) x F (x) = dx dx x x=1 0

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whenever F (0) = 0. Thus )  i ( Z 1 (−1)k d n+1 m (xy − 1) Vi,0 = y (x − 1) dx n+1 dy 0

361

. y=1

Therefore if i = 0 then we have Z (−1)k 1 1 V0,0 = (x − 1)k−1 dx = − = −W0 n+1 0 k(n + 1)

(11.41)

from Eq. (11.38). If i ≥ 1 then integrating by parts we get Z 1 (xy − 1)n+1 (x − 1)m dx 0

1 Z 1 m (xy − 1)n+2 m (x − 1) − (xy − 1)n+2 (x − 1)m−1 dx = y(n + 2) y(n + 2) 0 0

= ······



m 1 m!(n + 1)! − + · · · + (−1)m−1 m y(n + 2) y 2 (n + 2)(n + 3) y (m + n + 1)!  Z 1 m!(n + 1)! m m+n+1 +(−1) m (xy − 1) dx, y (m + n + 1)! 0 m m!(n + 1)! (y − 1)k X m!(n + 1)! = (−1)k+n + (−1)r+k+1 r+1 . m+1 k! y y (m − r)!(n + 2 + r)! r=0 = (−1)k+1

The substitution r → m − r yields Vi,0 = (−1) ( =

k+n m!n!

k!

(−1)

k+1

(−1)

k+1



d y dy

i (

(y − 1)k y m+1

)

y=1

W0

−

m X (−1)r+n m!n! r=0

r!(k − r)!

if 0 < i < k, k+n

W0 + (−1)

n!(k − n − 2)!

if i = k.

(r − 1 − m)i (11.42)

Thus when 0 < i < k and k is even we have Vi,0 = −Wi . It follows from Eq. (11.40), Eq. (11.41) and Eq. (11.42) that lim ζqR (−m, −n) = ζ R (s1 , s2 )

q→1

since Bk = 0 if k > 2 is odd. Let’s turn to prove the first equality in Eq. (11.37). Theorem 11.3.10 implies that 2  q−1 ζqI [−n, −m] = (q − 1)−k (A + C + D + E) log q

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E :=

∞ X Bi i=0

i!

(log q)i Ui ,

where Ui = Uim,n := −

n X

(−1)r+n

r=0

m!n! (n + 1 − r)i . r!(k − r)!

Hence U0m,n = W0n,m =

−1 1 = − D. k(m + 1) k(n + 1)

We only need to show that ( Wi , Ui = Wi − (−1)n n!(k − n − 2)!,

if 0 < i < k; if i = k.

(11.43)

(11.44)

Indeed when i > 0 we have ) i ( X    n d r m!n! k n+1−r n+1 y (−1) Ui = (−1) y dy k! r r=0 y=1 ) (    i k X d r k n+1−r n+1−k k n+1 m!n! (−1) y y y (y − 1) − = (−1) r k! dy r=n+1 y=1 ( )  i k k X m!n! d (y − 1) m!n! = (−1)n+1 y + (−1)r+n (n + 1 − r)i . k! dy y m+1 r!(k − r)! r=n+1 Then first term is 0 if i < k and it’s (−1)n+1 m!n! if i = k. When r = n + 1 the summand in the second term is 0 since i > 0. So we can let r range only from n + 2 to k. Then change the index r → k − r (and let r run from 0 to m) we can see immediately that the second term is exactly Wi . This proves Eq. (11.44) which together with Eq. (11.43) implies the first equation in Eq. (11.37). We thus finish the proof of the corollary. We conclude this section by remarking that by the stuffle relation Eq. (12.22) from Chap. 12 we can also analyze ζqI [s1 , s2 ] at (−n, n + 2 − k) for any nonnegative integers k and n. For example, it’s easy to compute directly that   3 X q r+1 1 I r 3 (−1) (r + 1) Res ζq [s1 , s2 ] = − (1 − q) log q r=0 r 1 − q r+1 (s1 ,s2 )=(2,−3) =

(q +

1)(q 2

−q(q − 1)2 , + 1)(q 2 + q + 1) log q

(11.45)

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which can be obtained also by the shuffle relation in Eq. (12.22) and the expression

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Res

(s1 ,s2 )=(−3,2)

ζqI [s1 , s2 ] =

q(q − 1)2 (q + 1)(q 2 + 1)(q 2 + q + 1) log q

by taking k = n = 3 in Thm. 11.3.3. Thus (2, −3) is a simple pole of the double q-zeta function ζqI [s1 , s2 ]. On the other hand the ordinary double zeta function ζ(s1 , s2 ) does not have a pole along s1 + s2 = −1. Indeed from Eq. (11.45) we find that  I lim Res ζq [s1 , s2 ] = 0. q→1 (s1 ,s2 )=(2,−3)

11.4 Historical Notes The q-analog of the Riemann zeta function ζ I [s] is first systematically studied from both analytic and algebraic points of view in [337]. In particular, the depth one case of Thms. 11.2.1 and 11.3.1 is proved there. A very interesting paper [341] by Kawagoe, Kurokawa and Wakayama investigated this q-Riemann zeta function along the same line but in more depth. In particular, both the behavior of zeros and the “crystal” case (q → 0) were considered in some details and an analog of the Riemann hypothesis was proposed. The general multiple q-zeta function ζ I [s] of arbitrary depth was treated first by the author in [615], mostly from the analytic point of view. Nonetheless, he also considered the special values at nonnegative integers when depth is two. In [337], Kawagoe, Kurokawa and Wakayama showed that if ϕ(s) is a ϕ(s) meromorphic function of s then the limit limq→1 ζq [s] = ζ(s) if and only if ϕ(s) = s − ν for some ν ∈ N. We don’t know in depth d > 1 case what (ϕ (s),...,ϕd (s)) one should expect of ϕj (s) if limq→1 ζq 1 [s] = ζ(s). Exercises 11.1. Prove Prop. 11.1.5 completely by using the mathematical induction and the binomial identity       n n n+1 + = . r r−1 r 11.2. Show by an appropriate substitution that Z ∞ x(s−1) q s−1 (1 − q)1−s q dx = − . x s (1 − q ) (s − 1) log q 1

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11.3. Prove Eq. (11.14) by repeatedly using integration by parts.

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11.4. Show that lim (1 − q)s log q

q→1

lim (1 − q)s log q

q→1

X n∈Z\{0}

X n∈Z\{0}

s bq (s − 1 + δn, −s) = 0, (2πin)2 1 bq (s − 1 + δn, −s + 1) = 0. (2πin)2

11.5. Let M > 2N + 16 and x, k > 0. Prove that  M −N +1 1−q x(M −N −2)−kM/6 g(q) = q 1 − qx is an increasing function for 1/2 < q < 1. 11.6. Prove Prop. 11.3.6 using Eq. (11.24). 11.7. Show that D0 and D1 satisfy the same bound as Eq. (11.22) if 1/2 < Pd q < 1 and M > 16 + 2N1 + 6 j=2 (Nj + 1) where the positive integers Nj > |Re(sj )| for all j. 11.8. Compute the residue Res

(s1 ,s2 )=(3,−4)

ζqI [s1 , s2 ].

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Chapter 12

q -Analogs of Multiple Zeta (Star) Values

There are quite a few different ways to define q-analogs of the MZVs and the MZ? Vs. We require that a good analog should not only deform into its classical counterpart when q → 1 but also satisfy the key relations that hold in the classical setting. Therefore, we seek for the q-analogs of the MZVs (q-MZVs for abbreviation) that obey some kind of double shuffle relations (DBSFs). On the other hand, for the q-analogs of the MZ? Vs (q-MZ? Vs for abbreviation), we choose those which enjoy interesting identities similar to the non-q version. In this chapter we present some of these and consider their most important properties. 12.1 Various Definitions of q-MZVs We have seen from last chapter that a good q-analog of the multiple zeta function is defined by X q k1 (s1 −1) · · · q kd (sd −1) . ζqI [s] := ζqs−1 [s] = [k1 ]s1 · · · [kd ]sd k1 >···>kd ≥1

We can define two more analogs as follows: ζqII [s] := ζqs [s] =

X k1 >···>kd ≥1

d Y q k1 s1 · · · q kd sd , [k1 ]s1 · · · [kd ]sd j=1

X

ζqIII [s] := ζq(1,0,...,0) [s] =

k1 >···>kd ≥1

q k1 . [k1 ]s1 · · · [kd ]sd

By Prop. 11.1.1 we obtain the following corollary immediately. Corollary 12.1.1. Let s = (s1 , . . . , sd ) ∈ Zd . Then (i) ζqI [s] converges if s1 + · · · + sj > j, for all j = 1, . . . , d. (ii) ζqII [s] converges if s1 + · · · + sj > 0, for all j = 1, . . . , d. (iii) ζqIII [s] always converges. 365

(12.1) (12.2)

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In this chapter we will also consider two more special types of the qMZVs that are more complicated than the above three types but have some advantages by themselves. Both of them have the form X k1 >···>kd ≥1

Ps1 (q k1 ) · · · Psd (q kd ) [k1 ]s1 · · · [kd ]sd

where Pn (t) are polynomials of t without constant term. The first class, called the Bachmann–K¨ uhn brackets (mono- and bi-brackets), uses the Eulerian polynomial P˜n (t) for Pn (t), where P˜n (t) =

s−1 X

Es,n tn ,

n=0

and the Eulerian numbers Es,n are defined by Es,n :=

  n X s+1 (−1)i (n + 1 − i)s . i i=0

The second class, called Okounkov’s q-MZVs and denoted by ζqO [s], is defined using the following palindrome polynomials: +

−

Pn (t) := tn + tn

(12.3)

where, for any n ∈ N, n− and n+ are the two nonnegative integers such that n n+1 n−1 ≤ n− ≤ ≤ n+ ≤ . 2 2 2 Clearly we have n+ + n− = n, n+ = n− if n is even, and n+ = n− + 1 if n is odd. All of the above q-MZVs can be expressed using the most general type (t ,...,td ) [s1 , . . . , sd ] defined by Eq. (11.1) where 1 ≤ t1 ≤ s1 , G q-MZVs ζq 1 0 ≤ tj ≤ sj for all j ≥ 2. They are all convergent by Prop. 11.1.1. 12.2 Double Shuffle Relations In this section, we will use a few algebras of words to encode the structure of various analogs of the q-MZVs. Motivated by the iterated Jackson qintegrals we can apply the theory of Rota–Baxter algebras to study the shuffle algebra. This enables us to obtain the DBSFs of these q-MZVs.

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Rota–Baxter Algebras

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We first consider some Rota–Baxter algebras which encode the (q-)shuffle relations for different versions of the q-MZVs. Definition 12.2.1. Fix an algebra A over a commutative ring R and an element λ ∈ R. We call A a Rota–Baxter R-algebra and P a Rota–Baxter operator of weight λ if the operator P satisfies the following Rota–Baxter relation of weight λ: P(x)P(y) = P(P(x)y) + P(xP(y)) + λP(xy) ∀x, y ∈ A.

(12.4)

Recall that for any continuous function f (x) on [α, β] the Jackson qintegral is defined by Z β X
f (x) dq x := f α + q k (β − α) (q k − q k+1 )(β − α). (12.5) α

k≥0

Taking α = 0 and β = t in Eq. (12.5) we now set X X   J[f ](t) := (1 − q) f (q k t)q k t = (1 − q) Ek I · f (t) k≥0

k≥0

= (1 − q)P[I · f ](t),

(12.6)

where I(t) = t is the identity function, E[f ](t) := Eq [f ](t) := f (qt), P[f ](t) := Pq [f ](t) := f (t) + f (qt) + f (q 2 t) + · · · are the q-expanding and the (principle) q-summation operators, respectively. We also need to define the (remainder ) q-summation operator R[f ](t) := Rq [f ](t) := f (qt) + f (q 2 t) + · · · = (P[f ] − [f ])(t). So, P is the principle part (i.e., the whole thing) while R is the remainder (i.e., without the first term). Clearly, P = R + I where, as an operator, I[f ] = f . Let tQ[[t, q]] be the space of formal series in two variables with t > 0. Then J, E, P and R are all Q[[q]]-linear endomorphism of tQ[[t, q]]. We can further define the inverse to P which is called the q-difference operator : D := I − E.

(12.7)

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Proposition 12.2.2. For any f, g ∈ tQ[[t, q]], we have     P[f ]P[g] = P P[f ]g + P f P[g] − P[f g],     R[f ]R[g] = R R[f ]g + R f R[g] + R[f g],     R[f ]P[g] = R R[f ]g + R f R[g] + R[f ]g + R[f g],       J[f ]J[g] = J J[f ]g + J f J[g] − (1 − q)J If g ,       = J f J[g] + qJ J E[f ] g ,

(12.8) (12.9) (12.10) (12.11) (12.12)

D[f ]D[g] = D[f ]g + f D[g] − D[f g],   D[f ]P[g] = D f P[g] + D[f ]g − f g,   D[f ]R[g] = D f R[g] + D[f g] − f g,

(12.13) (12.14) (12.15)

DP = PD = I.

(12.16)

Proof. For the first identity Eq. (12.8), we have 

 X X P[f ]P[g] (t) = f (q j t) g(q k t) j≥0

=

k≥0

X

f (q t)g(q k t) +

j≥k≥0

=

X

X

j

k≥j≥0

f (q

`+k

k

t)g(q t) +

`,k≥0

X

f (q j t)g(q k t) − j

f (q t)g(q

`+j

`,j≥0

X

f (q j t)g(q j t)

j≥0

t) − P[f g]

      = P P[f ]g + P f P[g] − P[f g] (t) All the other identities follow from the same type straightforward computation which is left as an exercise (see Exercise12.8). By Prop. 12.2.2 we see that P and R are both Rota–Baxter operators on tQ[[t, q]] (of weight −1 and 1, respectively) but D is not. In fact, D satisfies the condition Eq. (12.13) of a differential Rota–Baxter operator (see [259]). Moreover, it is invertible in the sense that Rota–Baxter operator P and the differential D are mutually inverse by Eq. (12.16). We end this section with an identity which will be used to derive the duality relations among the q-MZVs. For any n ∈ N, set Pn = P · · ◦ P} | ◦ ·{z n times

and Rn = R · · ◦ R} . | ◦ ·{z n times

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Theorem 12.2.3. Let d ∈ N and αj , βj ∈ N for all j = 1, . . . , d. Let t y(t) = 1−t . Then we have Rα1 yβ1 · · · Rα` yβ` (t) =

X

  `  Y jr − 1 kr − 1 βr − 1

j1 ≥β1 ,..., j` ≥β` r=1 k1 ≥α1 ,..., k` ≥α`

αr − 1

q

kr

P`

s=r

 t .

js jr

(12.17)

Proof. First we show that Rα (tj ) =

q αj tj . (1 − q j )α

(12.18)

Indeed, if α = 1 then R(tj ) =

X k≥1

q kj tj =

q j tj . 1 − qj

So Eq. (12.18) can be proved easily by the induction. Now we proceed to prove that for any integer m ≥ 0   Rα1 yβ1 · · · Rα` yβ` (t) · tm    `  X Y jr − 1 kr − 1 kr (m+P`s=r js ) jr m q t . (12.19) = t βr − 1 αr − 1 r=1 j1 ≥β1 ,..., j` ≥β` k1 ≥α1 ,..., k` ≥α`

If ` = 1 then we have     X β + j − 1 t β m Rα yβ (t) · tm = Rα t = Rα tm+β+j 1−t j j≥0 X j − 1 = Rα tm+j β−1 j≥β X  j − 1  q α(m+j) tm+j (by Eq. (12.18)) = β − 1 (1 − q m+j )α j≥β X  j − 1  X α + k − 1  = q (α+k)(m+j) tm+j β−1 k j≥β k≥0 X X  j − 1  k − 1  = q k(m+j) tm+j . β−1 α−1 j≥β k≥α

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This proves Eq. (12.19) when ` = 1. In general     Rα1 yβ1 · · · Rα`−1 yβ`−1 (t) Rα` yβ` (t) · tm =

  X  j` − 1  k` − 1  q k` (m+j` ) Rα1 yβ1 · · · Rα`−1 yβ`−1 (t) · tm+j` . β` − 1 α` − 1

j` ≥β` k` ≥α`

So Eq. (12.19) follows immediately by induction. We can now finish the proof of the theorem by taking m = 0. Corollary 12.2.4. Let d ∈ N and αj , βj ∈ N for all j = 1, . . . , d. Then we have Rα1 yβ1 · · · Rα` yβ` (1) = Rβ` yα` · · · Rβ1 yα1 (1).

(12.20)

Proof. In Eq. (12.17) we use the substitutions jr ↔ k`+1−r for all r = 1, . . . , `. Then we have ` ` X X r=1 s=r

js kr −→ =

` ` X X

j`+1−r k`+1−s =

r=1 s=r ` `+1−s X X

s ` X X

j`+1−r k`+1−s

s=1 r=1

j`+1−r ks =

s=1 r=1

` X ` X

jr ks =

s=1 r=s

` X

kr

r=1

` X

js

s=r

which follows from s ↔ ` + 1 − s followed by r ↔ ` + 1 − r and r ↔ s. This proves the corollary. 12.2.2

q-Analogs of Hoffman Algebras

We have seen that the DBSFs lead to many (and conjecturally all) Q-linear relations among the MZVs. To study similar relations of the q-MZVs we can modify the Hoffman algebra in this new setting. First we consider some algebras which will be used to define the stuffle relations later. Definition 12.2.5. Set the alphabet Xθ := {a, b, θ}. Denote by Aθ := Qha, b, θi the (weight) filtered noncommutative polynomial Q-algebra of words from Xθ∗ . The weights of a, b and θ are all equal to one. Set γ := b − θ,

zs := as−1 b,

z0s := as−1 θ,

s ∈ Z.

Let Y˜I := {θ} ∪ {zk }k≥1 , YII := {z0k }k≥0 , and YIII := {zk }k∈Z . We put a tilde on top of I since we need to consider some kind of regularization due to convergence issues involved in type I q-MZVs. This is realized by the

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371

introduction of the letter θ. Again, we use 1 for the empty word and Yτ∗ to denote the set of words generated on Yτ for any type τ . Let A˜1I , A1II and AIII be the subalgebras of Aθ freely generated by the sets Y˜I, YII and YIII , respectively. Set X A1III = z0k AIII 6⊂ AIII . k∈Z

Here, all integer subscripts are allowed in YIII because type III q-MZVs converge for all integer arguments. Further, we define the following subalgebras corresponding to the convergent values: X X A0I := Q1 + zk A˜1I , A˜0I := Q1 + θA˜1I + zk A˜1I ( A˜1I , k≥2

A0II

:= Q1 +

X

k≥2

z0k A1II

(

A1II ,

A0III

:=

A1III .

k≥1

For each type τ , the words in A0τ are called type τ -admissible. It is consistent with Prop. 11.1.1 since we consider only nonnegative compositions s. To define the stuffle product for type τ = ˜I and II, similar to the MZV case, we define a commutative product [−, −]τ first: [zk , zl ]˜I := zk+l + zk+l−1 , [θ, zk ]˜I := zk+1 ,

[z0k , z0l ]II := z0k+l ,

[θ, θ]˜I := z2 − θ,

(12.21)

for all k, l ≥ 1. Now we define the stuffle product ∗τ on A1τ inductively as follows. For any words u, v ∈ A1τ and letters α, β ∈ Yτ , we set 1 ∗τ u = u = u ∗τ 1 and (αu) ∗τ (βv) = α(u ∗τ βv) + β(αu ∗τ v) + [α, β]τ (u ∗ v).

(12.22)

Remark 12.2.6. It is not hard to check that for τ = ˜I and II, (A0τ , ∗τ ) ⊂ (A1τ , ∗τ ) as subalgebras. Definition 12.2.7. Define the injective shifting operator S− on any word of A0III by acting on the first letter: S− (z0n w) := zn w − zn−1 w

∗ for all n ∈ Z and w ∈ YIII .

(12.23)

∗ For any k, l ∈ Z and any u, v ∈ YIII , define the stuffle product ∗III by

0 0 0 0 0 zk u ∗III zl v = zk u ∗II S− (zl v) + zl S− (z0k u) ∗II v + (z0k+l − z0k+l−1 )(u ∗II v).

Here ∗II is the ordinary stuffle with [zr , zs ]II = zr+s for all r, s ∈ Z.

Lemma 12.2.8. Let τ = ˜I, II or III. Then the stuffle products ∗τ is well defined. Namely, u ∗τ v ∈ A1τ if u, v ∈ A1τ .

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Proof. The computation is straightforward. See Exercise 12.2.

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Proposition 12.2.9. Let τ = ˜I, II or III. Then the stuffle algebras (A1τ , ∗τ ) are all commutative and associative. Proof. This follows from the fact that the product [−, −]τ are all commutative and associative which can be verified easily. We now turn to the shuffle algebra which is an analog of the corresponding algebra for MZVs reflecting the properties of their representations using iterated integrals. Definition 12.2.10. Let Xπ = {π, δ, y} be an alphabet and Xπ∗ be the set of words generated by Xπ . Define Aπ := Qhπ, δ, yi to be the noncommutative polynomial Q-algebra of words of Xπ∗ . We may embed Aθ defined by Definition 12.2.5 as a subalgebra of Aπ in two different ways: put ρ = π − 1 and let (A) a := π, (B) a := ρ,

a−1 := δ, −1

a

:= −,

b := πy,

θ = ρy

=⇒

γ := y,

b := πy,

θ = ρy

=⇒

γ := y.

(A)

(B)

We denote the image of the embedding by Aθ and Aθ , respectively. The dash − for the image of a−1 in (B) means it does not matter what image we choose since a−1 appears only when we consider type III q-MZVs using (A). We will use embedding (B) for the other types for which a−1 will not be utilized essentially because of convergence issues. 12.2.3

q-Stuffle Relations

We now apply the results from the preceding section to study the stuffle relations of the q-MZVs. Define the Q-linear realization maps zq : A0τ → C (τ = ˜I, II) by zq [1] = 1 and X zq [y1τ . . . ydτ ] := Mkτ1 (y1τ ) . . . Mkτd (ydτ ), k1 >···>kd >0

where y1τ . . . ydτ ∈ A0τ and the Q-linear maps ˜

MkI (θ) := ˜

qk , (1 − q k )

˜

MkI (zs ) :=

q (s−1)k , (1 − q k )s

MkII (z0s ) :=

q sk . (1 − q k )s

˜

Note that MkI (γ) = MkI (z1 − θ) = 1. For example, we have zq [z2 z5 γ 2 z1 ] = z(1,4,0,0,0) [2, 5, 0, 0, 1], q

zq [θz7 θz4 ] = zq(1,6,1,3) [1, 7, 1, 4],

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373

which are not q-MZVs of type I. For type III, we similarly define the Q-linear realization maps zq : A0III → C by zq [1] = 1 and X zq [y1 . . . yd ] := Mk1,III (y1 )MkIII2 (y2 ) . . . MkIIId (yd ), 1 k1 >···>kd >0

where y1 . . . yd ∈

A0III

and the Q-linear maps

Mk1,III (z0s ) :=

qk , (1 − q k )s

MkIII (zs ) :=

1 . (1 − q k )s

Theorem 12.2.11. Let τ = ˜I, II or III. For any uτ , vτ ∈ A0τ , we have zq [uτ ∗τ vτ ] = zq [uτ ]zq [vτ ].

(12.24)

Proof. Let’s consider type ˜I first. For a positive integer N , we define the Q-linear map FN : A1I → C by FN (1) = 1 and X ˜ ˜ FN (α1 . . . αd ) = MkI1 (α1 ) . . . MkId (αd ), (12.25) N >k1 >···>kd >0

P ˜I where αj ∈ Y˜I for all j = 1, . . . , d. Since FN (αu) = N >m>0 Mm (α)Fm (u) for any α ∈ Y˜I and u ∈ A1θ . we see that zIq [u] = limN →∞ FN (u). Thus we only need to show that FN (u ∗ v) = FN (u)FN (v)

(12.26)

for all words u, v in Yθ not starting with z1 . Indeed, Eq. (12.26) can be proved by using the induction on |u| + |v|. Note that Eq. (12.26) is trivial if u or v is empty word 1. So we assume u, v ∈ A0θ and α, β ∈ Yθ \ {z1 }. Then X  ˜ ˜I I FN (αu)FN (βv) = Mm (α)Fm (u)Fm (βv) + Mm (β)Fm (αu)Fm (v) N >m>0

 ˜I ˜I (β)Fm (u)Fm (v) + Mm (α)Mm X  ˜ ˜I I (β)Fm (αu ∗ v) = Mm (α)Fm (u ∗ βv) + Mm N >m>0

 ˜I ˜I + Mm (α)Mm (β)Fm (u ∗ v) by inductive assumption. Hence Eq. (12.26) follows from Eq. (12.21) and ˜

˜

˜

˜

I I I I Mm (zk )Mm (zl ) = Mm (zk+l ) + Mm (zk+l−1 ), ˜I ˜I Mm (θ)Mm (zk )

˜I = Mm (zk+1 ),

(12.27) (12.28)

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Multiple Zeta Functions, Multiple Polylogarithms, and Their Special Values ˜

˜

˜

I I I Mm (θ)2 = Mm (z2 ) − Mm (θ)

(12.29)

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for all k, l ≥ 2 and m ≥ 1. The same argument works for type II and III if we observe that II II II Mm (z0k )Mm (z0l ) = Mm (z0k+l ), III III III Mm (zk )Mm (zl ) = Mm (zk+l ),

1,III 0 III III Mm (zk )Mm (zl ) = Mm (zk+l − zk+l−1 ),

1,III 0 1,III 0 1,III 0 Mm (zk )Mm (zl ) = Mm (zk+l − z0k+l−1 ),

for all k, l ≥ 0, m ≥ 1. This completes the proof of the theorem. 12.2.4

Iterated Jackson q-Integrals

As we have seen in section 3.2, the iterated integral representations of the MZVs and its shuffle product structure is the driving force in defining Hoffman’s shuffle algebra of words. Set x0 := x0 (t) =

1 , t

x1 := x1 (t) =

1 , 1−t

y := y(t) =

t . 1−t

Recall that for a = x0 (t) dt and b = x1 (t) dt, we can express MZVs by Chen’s iterated integrals: Z 1 ζ(s1 , . . . , sd ) = as1 −1 b · · · asd −1 b. 0

In the current q-analog setting, replacing the Riemann integrals by the Jackson q-integrals in Eq. (12.6) one gets the following two generalizations after slight modification. First, for any (a1 , . . . , ad ), (b1 , . . . , bd ) ∈ (Z≥0 )d , we define zq [ρa1 π b1 y . . . ρad π bd y; t]   := Ra1 Pb1 [yRa2 [Pb2 [y · · · Rad [Pbd [y]] · · · ]]] (t). (12.30) Theorem 12.2.12. Let s = (s1 , . . . , sd ) ∈ Nd and a = (a1 , . . . , ad ) ∈ (Z≥0 )d . Put w = |s| and w = wa (s) = ρa1 π s1 −a1 y . . . ρad π sd −ad y. If a1 + · · · + aj > 0 for all j = 1, . . . , d, then we have ζqa [s] = (1 − q)w zq [wa (s); 1],

zaq [s] = zq [wa (s)] := zq [wa (s); 1]. (12.31)

Proof. First we observe three important facts: for any k ≥ 1 we have P(tk ) =

X j≥0

q kj tk =

tk , 1 − qk

R(tk ) =

X j≥1

q kj tk =

q k tk 1 − qk

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375

and D(tk ) = tk (1 − q k ), by the definition of the two summation operators and the difference operator. Repeatedly applying this we get X tk Pm (tk ) = q kj tk = ∀m ∈ Z, (12.32) (1 − q k )m j≥0

Rm (tk ) =

X j≥1

q kj tk =

q mk tk (1 − q k )m

∀m ∈ Z≥0 .

(12.33)

Thus XX X t`
X q j(k+1) tk+1 j(k+`+1) k+`+1 . P y(t) · tk = = q t = 1 − qj t 1 − q` j≥0

j≥0 `≥0

`>k

Similarly, we have X X
tk+1 q k+1 tk+1 D y(t) · tk = − = (1 − q k+`+1 )tk+`+1 = (1 − q ` )t` , 1−t 1 − qt `≥0

`>k

and XX X q ` t`
X q j(k+1) tk+1 j(k+`+1) k+`+1 R y(t) · tk = = q t = . 1 − qj t 1 − q` j≥1

j≥1 `≥0

`>k

It follows from Eq. (12.32) and Eq. (12.33) that
X t` ∀m ∈ Z, Pm y(t) · tk = (1 − q ` )m

(12.34)

`>k


X Rm y(t) · tk = `>k

q m` t` (1 − q ` )m

∀m ∈ Z≥0 .

(12.35)

We now prove by induction on d that for all s = (s1 , . . . , sd ) ∈ Nd , X tk1 q k1 a1 . . . q kd ad zq [wa (s); t] = . (12.36) (1 − q k1 )s1 · · · (1 − q kd )sd k1 >···>kd >0

When d = 1, i.e., s = s, then by Eqs. (12.34) and (12.33)
X Ra tk X q ak tk a a s−a zq [w (s); t] = R P [y](t) = = = z(a) q [s; t]. (1 − q k )s−a (1 − q k )s k>0

k>0

This proof works even when s = a because of Eq. (12.35) (take k = 0 and m = a there). In general, assume d ≥ 2 and Eq. (12.36) is true for smaller depths. Then by the inductive assumption   zq [wa (s); t] = Ra1 Ps1 −a1 yRa2 Ps2 −a2 [y · · · Rad Psd −ad [y] · · · ] (t)

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Multiple Zeta Functions, Multiple Polylogarithms, and Their Special Values


Ra1 Ps1 −a1 y(t) · tk2 q k2 a2 . . . q kd ad = (1 − q k2 )s2 · · · (1 − q kd )sd k2 >···>kd >0
X Ra1 tk1 q k2 a2 . . . q kd ad (12.34) == (1 − q k1 )s1 −a1 (1 − q k2 )s2 · · · (1 − q kd )sd

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X

k1 >···>kd >0

(12.33)

X

==

k1 >···>kd >0

tk1 q k1 a1 . . . q kd ad . (1 − q k1 )s1 · · · (1 − q kd )sd

Again, if s1 = a1 the proof is still valid. This completes the proof of Eq. (12.36). Setting t = 1 we arrive at Eq. (12.31). By the change of variables aj → sj − aj for all j = 1, . . . , d, we immediately obtain the next result. Theorem 12.2.13. For s = (s1 , . . . , sd ) ∈ Nd and a = (a1 , . . . , ad ) ∈ (Z≥0 )d , we set s − a = (s1 − a1 , . . . , sd − ad ), w = |s| and ws−a (s) = ρs1 −a1 π a1 y . . . ρsd −ad π ad y. Define   zq [ws−a (s); t] := Rs1 −a1 Pa1 [yRs2 −a2 [Pa2 [y · · · Rsd −ad [Pad [y]] · · · ]]] (t). If s1 + · · · + sj > a1 + · · · + aj for all j = 1, . . . , d, then we have ζqs−a [s] = (1 − q)w zq [ws−a (s); 1],

(12.37)

zs−a [s] = zq [ws−a (s)] := zq [ws−a (s); 1]. q

By specializing the preceding two theorems we quickly find the following corollary. For future reference, we will say wτ has the typical type τ form for each type τ . Corollary 12.2.14. For s = (s1 , . . . , sd ) ∈ Nd , we set (B)

wI = ρs1 −1 πy . . . ρsd −1 πy = zs1 . . . zsd ∈ Aθ (B)

wII = ρs1 y . . . ρsd y = z0s1 . . . z0sd ∈ Aθ wIII = π

s1 −1

s2

sd

ρyπ y . . . π y =

z0s1 zs2

⊂ Aπ y,

⊂ Aπ y (A)

. . . zsd ∈ Aθ

(s1 ≥ 2),

⊂ Aπ y,

and   zq [wI ; t] := Rs1 −1 P[yRs2 −1 [P[y · · · Rsd −1 [P[y]] · · · ]]] (t),   zq [wII ; t] := Rs1 yRs2 [y · · · Rsd [y] · · · ] (t), h  i zq [wIII ; t] := Ps1 −1 R y[Ps2 [y[Ps3 [y · · · Psd [y] · · · ]]]] (t). Then for all the types τ = I, II and III, we have ζqτ [s] = (1 − q)w zq [wτ ; 1],

zτq [s] = zq [wτ ] := zq [wτ ; 1].

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Moreover, similar results hold for type ˜I q-MZVs. We may replace any of the consecutive strings ρsj −1 π by a single ρ in w˜I, and replace the corresponding operator string Psj −1 P by a single R. We now apply the above to Okounkov’s q-MZVs. Let s ∈ (Z≥2 )d . Then using Eq. (12.3) we define X

ζqO [s] :=

k1 >···>kd >0

= (1 − q)|s|

+ − d Y q kj sj + q kj sj [kj ]sj j=1

X k1 >···>kd >0

+ − d Y q kj sj + q kj sj . (1 − q kj )sj j=1

Again, its modified form is: zO q [s]

X

:=

k1 >···>kd >0

+ − d Y q kj sj + q kj sj . (1 − q kj )sj j=1

Corollary 12.2.15. For s = (s1 , . . . , sd ) ∈ Nd , we set −

+

+

−

−

+

+

−

(B)

wO = (ρs1 π s1 + ρs1 π s1 )y . . . (ρsd π sd + ρsd π sd )y ∈ Aθ

⊂ Aπ y

and  − + + −  − + + − zq [wO ; t] = (Rs1 Ps1 + Rs1 Ps1 ) y · · · (Rsd Psd + Rsd Psd )[y] · · · (t). Then we have ζqO [s] = (1 − q)w zq [wO ; 1],

zO q [s] = zq [wO ] := zq [wO ; 1].

It is possible to obtain the shuffle relations among zO q [s]-values using Cor. 12.2.15. The stuffle relations among zO q [s] is mentioned implicitly in Okounkov’s original paper. For our modified version, they can be derived + − from the following fact. Let FnO (t) = (tn + tn )/(1 − t)n for all n ≥ 2. Then for all r, s ∈ Z≥2 , we have  O 2Fr+s (t), if r or s is even; FrO (t) · FsO (t) = O O 2Fr+s (t) + 12 Fr+s−2 (t), if r and s are odd. For example, O O O O O zO q [2, 3]zq [2] = 2zq [2, 2, 3] + zq [2, 3, 2] + 2zq [4, 3] + 2zq [2, 5],

1 O O O O O O zO q [2, 3]zq [3] = 2zq [2, 3, 3] + zq [3, 2, 3] + 2zq [5, 3] + 2zq [2, 6] + zq [2, 4]. 2

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12.2.5

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q-Shuffle Relations

In contrast to the MZV case, the q-shuffle product is much more difficult to define than the q-stuffle product. In this section we will use the Rota– Baxter algebra approach to define this for type ˜I, II, and III q-MZVs. The q-shuffle product on Aπ is defined recursively as follows: for any words u, v ∈ Xπ∗ we define 1 u = u 1 = u and








v = u
(yv) = y(u
v), πu
πv = π(u
πv) + π(πu
v) − π(u
v), δu
δv = u
δv + δu
v − δ(u
v), δu
πv = πv
δu = δ(u
πv) + δu
v − u
v,

(yu)

(12.38) (12.39) (12.40) (12.41)

for any words u, v ∈ Xπ∗ . The first equation reflects the fact that when y(t) is multiplied in front of either of the two factors in a product, it can be multiplied after taking the product. The other equations formalize Eq. (12.9),Eq. (12.8), Eq. (12.13), Eq. (12.14), and Eq. (12.15), respectively. Corollary 12.2.16. For any words u, v ∈ Xπ∗ , we have


ρv = ρ(u
ρv) + ρ(ρu
v) + ρ(u
v),
πv = πv
ρu = ρ(ρu
v) + ρ(u
ρv) + ρu
v + ρ(u
v), δu
ρv = ρv
δu = δ(u
πv) − u
v = δ(u
ρv) + δ(u
v) − u
v.

ρu

(12.42)

ρu

(12.43) (12.44)

Proof. These follow easily from Eq. (12.38)–Eq. (12.41) and the relation ρ = π − 1. (j)

(j),∗

Corollary 12.2.17. For j = 1, 2, let Xθ and Xθ be the embedding of (j) Xθ and Xθ∗ into Xπ∗ , respectively, by Definition 12.2.10. For any α, β ∈ Xθ (j),∗ and u, v ∈ Xθ , we have 1 u = u 1 = u and




αu





βv = α(u
βv) + β(αu
v) + [α, β] (u
v), j

where [α, β]j is determined by [a, b]1 = [b, a]1 = −b, [a, b]2 = [b, a]2 = 0 and [a, a]j = (−1)j a,

[b, b]j = −bγ,

[α, γ]j = [γ, α]j = −αγ.

(12.45)

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Proof. All of these identities follow from straightforward computation using Eqs. (12.38)–(12.44). For example,

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bu


bv = πyu
πyv = π(yu
πyv) + π(πyu
yv) − π(yu
yv) = πy(u
πyv) + πy(πyu
v) − πyy(u
v) = b(u
bv) + b(bu
v) − bγ(u
v).

(12.46)


θv = ρyu
ρyv = ρ(yu
ρyv) + ρ(ρyu
yv) + ρ(yu
yv) = ρy(u
ρyv) + ρy(ρyu
v) + ρyy(u
v) = θ(u
θv) + θ(θu
v) + θγ(u
v).

(12.47)

Similarly, θu

The rest of the proof is left to the interested reader. Proposition 12.2.18. The algebra (Aπ , tive.


) is commutative and associa-

Proof. Proving commutativity is routine and is thus left to the interested reader as an exercise. The associativity relation (u v) w = u (v w) is much harder. We now prove it by induction on |u| + |v| + |w|. Without loss of generality we write u = αa, v = βb, and w = γc where α, β, γ ∈ {π, δ, y} and a, b, c ∈ Xπ∗ . The 27 different cases can all be reduced to the following 5 cases by commutativity:









(1) One of the letters α, β or γ is a y. By inductive assumption and Eq. (12.38):

(ya v) u = y(a v) u = y (a v) u
= y a (v u) = ya (v u).

















Hence associativity in this case is proved. (2) One π and two δ’s. By commutativity we may assume α = β = δ and γ = π. By Eq. (12.40) and Eq. (12.41)


δb)
πc
= a
δb + δa
b − δ(a
b)
πc = a
δb
πc + b
δa
πc − δ(a
b)
πc = a
δ(b
πc) + a
δb
c − a
b
c + b
δ(a
πc) + b
δa
c − b
a
c − δ(a
b
πc) − δ(a
b)
c + a
b
c

(δa

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Multiple Zeta Functions, Multiple Polylogarithms, and Their Special Values


δ(b{z
πc)} + a|
δb b
c} + b
δ(a
πc) {z
c} − a |
{z | {z }

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1

2

5

6

3

4

b
πc) − δ(a
b)
c .
δa {z
c} − |δ(a
{z } | {z } 7

On the other hand, δa


(δb
πc) = δa
δ(b
πc) + δb
c − b
c) = δa
δ(b
πc) + δa
δb
c − δa
b
c = a
δ(b
πc) + δa
πc
b − δ(a
b
πc) + a
δb
c + δa
b
c − δ(a
b)
c − δa
b
c c
b} − a c
b} = a
δ(b
πc) + δ(a
πc)
b + |δa
{z |
{z {z } | {z } | 1

− δ(a |

4

5

2

7


{zb
πc)} + a|
δb
{zb)
c} . {z
c} − δ(a | 6

3

Hence associativity in case (2) is proved. (3) Two π’s and one δ. By commutativity we may assume α = β = π and γ = δ. By Eq. (12.39) and Eq. (12.41)


πb)
δc = π(a
πb)
δc + π(πa
b)
δc − π(a
b)
δc
= δ c
π(a
πb) + δc
a
πb − a
πb
c
+ δ c
π(b
πa) + δc
b
πa − b
πa
c
− δ (c
π(a
b) − δc
a
b + c
a
b
= δ c
π(a
πb) + a
δ(c
πb) + a
δc
b − a
c
b
− a
πb
c + δ c
π(b
πa) + b
δ(c
πa) + b
δc
a
− b
c
a − b
πa
c − δ (c
π(a
b) − δc
a
b + c
a
b.
= δ c
π(a
πb) + a
δ(c
πb) − a
c
b {z } | {z } | {z } |

(πa

3

2

1

− |a


πb
πa)} + b|
δ(c{z
πa)} {z
c} + δ| c
π(b {z

+b |



δc
b) . {z
a} − b |
πa {z
c} − δ| c
π(a {z }




4

7

8

On the other hand: πa

6

5


(πb
δc)

9

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381


δ(c
πb) + δc
b − c
b
= δ(c
πb
πa) + δ(c
πb)
a − c
πb
a + δ(c
πa)
b + δc
a
b − a
c
b − πa
c
b


= δ (c
π(b
πa) + δ c
π(πb
a) − δ c
π(b
a) | {z } | {z } | {z }

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= πa

5

+ δ(c |

1


{zπb)
a} − c|
πb
{zπa)
b} {z
a} + δ(c | 4

2

9

6



b} − a|
{zc
b} − πa |
{zc
b} .

+ |δc {z a 7

3

8

Hence associativity in case (3) is proved. (4) α = β = γ = δ. By inductive assumption, commutativity, and Eq. (12.40)


δb)
δc
= a
δb + δa
b − δ(a
b)
δc = a
δb
δc + b
δa
δc − δ(a
b)
δc
= a
b
δc + δb
c − δ(b
c) − a
b
δc
+ b
a
δc + δa
c − δ(a
c) − δ(a
b)
c + δ(a
b
c) = |a
δb
c)} + b|
a{z
δc} + b|
δa {z
c} − |a
δ(b {z
c} {z

(δa

1

−b |

3

2


δ(a
c)} − δ(a
{zb)
c} + δ(a
{zb
c)} . {z | | 5

6

4

7

On the other hand,


(δb
δc)
= δa
b
δc + δb
c − δ(b
c) = δa
δc
b + δa
δb
c − δa
δ(b
c) = a
δc
b + δa
c
b − δ(a
c)
b + a
δb
c + δa
b
c − δ(a
b)
c − a
δ(b
c) − δa
b
c + δ(a
b
c). = |a
δc c)
b + a
δb b
c} {z
b} − |δ(a
{z {z
c} + δa |
{z } |

δa

3

1

5



c} − |a
δ(b
c)} + δ(a
{zb
c)} . {z |

− δ(a b) | {z 6

2

7

Hence associativity in case (4) is proved.

4

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(5) α = β = γ = π. This is the ordinary stuffle case. We leave it as an exercise.

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This concludes the proof of the proposition. (j)

Corollary 12.2.19. For j = 1 or 2, the algebras (Aθ , and associative.


) are commutative
) are sub-

(j)

Proof. This follows immediately from Prop. 12.2.18 since (Aθ , (j) algebras of (Aπ , ) if for Aθ is defined as in Cor. 12.2.17.





Our next theorem shows that we may use the shuffle algebra structure defined above to describe the shuffle relations among different types of the q-MZVs. Before doing so, we need to show that for each type the shuffle product really makes sense. (B)

(A)

Proposition 12.2.20. Embed A˜0I , A0II ⊂ Aθ and A0III ⊂ Aθ . Then for each type τ , if the two words u, v ∈ A0τ have the typical type τ form listed in Cor. 12.2.14 then there is an algorithm to express u v using only those words in the same form.




Proof. For type ˜I, we prove in general that for k, l ≥ 1 and u, v ∈ A1I X zk u zl v ∈ zj A1θ . (12.48)




j≥min (k,l)

If k = l = 1 Eq. (12.48) follows from b = z1 and α(b, b) = −bγ. If k = 1 and l ≥ 2, we can prove Eq. (12.48) by induction on l using bu


z v = b(u
z v) + a(bu
z l

l−1 v)

l

since azj = zj+1 for all j ≥ 1. Now we assume k, l ≥ 2. Then zk u


z v = a(z

k−1 u

l

∈


z v+z u
z l

X

k

l−1 v

+ zk−1 u


z

l−1 v)

azj−1 A˜1I

j≥min (k,l)

by induction assumption. Thus Eq. (12.48) holds. Finally we prove that θu


z v, θu
θv ∈ A

0 ˜I

k

for all u, v ∈ A˜1I and k ≥ 2. This follows easily from Eq. (12.48) and


z v = θ(u
z v) + a(bu
z v),
θu
θv = θ u
θv + θu
v − γ(u
v) .

θu

k

k

k−1

(12.49)

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383

Type II is in fact the easiest since we can restrict ourselves to use only Eq. (12.38) and Eq. (12.42) to compute the shuffle and therefore π never comes into the picture. Clearly all such words must start with ρ and end with y. For type III, let’s assume u = π s1 −1 ρyπ s2 y . . . π sd y and v = π t1 −1 ρyπ t2 y . . . π td y. If we use the definition Eq. (12.39) repeatedly then in each word appearing in u v the first ρ always appears before all the y’s. Such a word can be written in the form π s ρr y · · · for some s ∈ Z and r ≥ 1 (notice that if ρ and π are commutative). Now we can rewrite this as π s (π − 1)r−1 ρy · · · and replace all the ρ’s after the first y by π − 1. This produces a word of typical type III form. We have completed the proof of the proposition.




(B)

Theorem 12.2.21. Embed A˜0I , A0II ⊂ Aθ type τ and for any uτ , vτ ∈ A0τ , we have

zq [uτ ]zq [vτ ] = zq [uτ

(A)

and A0III ⊂ Aθ . Then for each


v ]. τ

(12.50)

Proof. For each type τ , we observe that zq [wτ ; t] satisfies Eq. (12.50) because of the identities in Prop. 12.2.2. Then the theorem follows from the fact that zq [wτ ] = zq [wτ ; 1] for any word wτ ∈ A0τ by Cor. 12.2.14. 12.3 Duality Relations The DBSFs do not contain all linear relations among the various types of the q-MZVs. Some of the missing ones can be proved by the following “duality” relations because of their similarity to the duality relations for the ordinary MZVs. Theorem 12.3.1. For a positive integer k, set   k X ϕk := (−1)k  (−1)j zj − θ , j=2

(B)

where ϕ1 = θ = ρy ∈ Aθ . Let ` ∈ N and αj , βj ∈ Z≥0 for all j = 1, . . . , `. Then we have ˜

˜

ζqI [ϕα1 +1 γ β1 · · · ϕα` +1 γ β` ] = ζqI [ϕβ` +1 γ α` · · · ϕβ1 +1 γ α1 ].

(12.51)

Proof. Note that γ = y, zj = ρj−1 πy and θ = ρy with the embedding

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(B)

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A˜0I ⊂ Aθ . Since π = ρ + 1, for all k ≥ 1, we have   k X ϕk =(−1)k  (−1)j ρj−1 (ρ + 1)y − ρy  j=2

  k k X X =(−1)k  (−1)j ρj y + (−1)j ρj−1 y − ρy  = ρk y. j=2

(12.52)

j=2

Thus the theorem follows from Cor. 12.2.4 and Cor. 12.2.14. We see that in the expression of ϕk the letter θ appears, however, qMZVs of the form such as ζqI [θγz2 γ] = ζqI [ρy 2 ρ2 πy] is not really defined. In ˜

(1,0,1,0)

[1, 0, 2, 0] (and such values fact, it should be denoted by ζqI [θγz2 γ] = ζq always converge by Prop. 11.1.1 because of the leading 1 in the auxiliary variable t). But, suitable Q-linear combinations of Eq. (12.51) may lead to identities in which only zk ’s appear. Then all terms can be written as honest ζqI -values. This explains the use of two admissible structures A0I and A˜0I . Similar relations for type II q-MZVs have the most aesthetic appeal and is the primary reason why we prefer to call it by “duality”. Theorem 12.3.2. Let ` ∈ N and αj , βj ∈ N for all j = 1, . . . , `. Then we have ζqII [ρα1 y β1 · · · ρα` y β` ] = ζqII [ρβ` y α` · · · ρβ1 y α1 ]. Proof. This follows from Cor. 12.2.4 and Cor. 12.2.14 immediately. Of course we may apply the same idea to type III q-MZVs. Theorem 12.3.3. Let ` ∈ N and αj , βj ∈ N for all j = 1, . . . , `. Then we have ζqIII [(π − 1)α1 −1 ρy β1 (π − 1)α2 y β2 · · · (π − 1)α` y β` ]

= ζqIII [(π − 1)β` −1 ρy α` (π − 1)β`−1 y α`−1 · · · (π − 1)β1 y α1 ].

Proof. Since ρ = π − 1 the theorem follows readily from Cor. 12.2.4 and Cor. 12.2.14.

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12.4 The P-R and R-P Relations We observe that there is often more than one way to express a q-MZV using words because of the relation π = ρ + 1. For example, replacing π by ρ + 1 we have π 2 ρy = πρ2 y + πρy = ρ3 y + 2ρ2 y + ρy which yields immediately the relations (2) (1) (3) (2) (1) z(1) q [3] = zq [3] + zq [2] = zq [3] + 2zq [2] + zq [1].

We call all such relations P-R relations (pronounced as “P to R relations”). On the level of the q-MZVs this is a consequence of the following identity q nt q nt − q (n+1)t + q (n+1)t q nt q (n+1)t = = + . (1 − q n )s (1 − q n )s (1 − q n )s−1 (1 − q n )s (t ,...,t )

d [s1 , . . . , sd ] defined Now we consider the general type G q-MZVs ζq 1 by Eq. (11.1) where 1 ≤ t1 ≤ s1 , 0 ≤ tj ≤ sj for all j ≥ 2. Repeatedly applications of the P-R relation shows that all these values can be reduced to type II q-MZVs since in the word expression we only need to use ρ and y. On the other hand, if we use ρ by π − 1 repeatedly we may convert (t ,...,td ) [s1 , . . . , sd ] to type III values. We call these all type G q-MZVs ζq 1 relations R-P relations (pronounced as “R to P relations”). For example, from

ρ3 y = (π 2 − 2π + 1)ρy we get ζqII [3] = ζqIII [3] − 2ζqIII [2] + ζqIII [1]. 12.5 General Type G q-MZVs All of the q-MZVs of type ˜I, II and III considered earlier in the chapter are (t ,...,td ) some special forms of the q-MZVs ζq 1 [s1 , . . . , sd ] defined by Eq. (11.1) where 1 ≤ t1 ≤ s1 , 0 ≤ tj ≤ sj for all j ≥ 2, all of which are convergent by Prop. 11.1.1. Let A0G be the set of all such values which we call (admissible) type G q-MZVs. Similar to the above three types, we may use words to encode these values according to Thm. 12.2.12 by setting aj = tj there. Namely, setting wt (s) = ρt1 π s1 −t1 y · · · ρtd π sd −td y ∈ Xπ∗ ,

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we have

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zq(t1 ,...,td ) [s1 , . . . , sd ] = zq [wt (s)] where the right-hand side is defined by Eq. (12.30). The shuffle product structure is reflected by (Xπ∗ , ) where the is defined by Eqs. (12.38), (12.39), (12.42) and (12.43).







Proposition 12.5.1. For all u, v ∈ A0G , we have u


v ∈A

0 G.

Proof. Note that admissible words in A0G must end with y and have at least one ρ before the first y. Moreover, the converse is also true. This is rather straightforward if we use the P-R relations repeatedly to get rid of all the π’s. Now, by using the definition of it is not hard to see that u v ends with y and has at least one ρ before the first y if both u and v are admissible. So u v ∈ A0G and the proposition is proved.










To define the stuffle product, we set YG := {zt,s | t, s ∈ Z≥0 , t ≤ s}, ∗ and let AG be the noncommutative polynomial Q-algebra of words of YG built on the alphabet YG . Define the type G-admissible words as those in [ A0G := zt,s AG . 1≤t≤s

We can regard AG as a subalgebra of Xπ∗ by setting zt,s = ρt π s−t y. Then stuffle product ∗G on A0G can be defined inductively as follows. For any words u, v ∈ A0G and letters zt,s , zt0 ,s0 ∈ YG with 1 ≤ t ≤ s and 1 ≤ t0 ≤ s0 , we set 1 ∗G u = u = u ∗G 1 and (zt,s u) ∗G (zt0 ,s0 v) = zt,s (u ∗G zt0 ,s0 v) + zt0 ,s0 (zt,s u ∗G v) + zt+t0 ,s+s0 (u ∗G v). It is easy to show that (A0G , ∗G ) is a commutative and associative algebra. We leave the proof of the following theorems to the interested reader. The first result clearly provides the DBSFs of type G q-MZVs. Theorem 12.5.2. For any u, v ∈ A0G ⊂ Xπ∗ , we have zq [u ∗G v] = zq [u


v] = z [u]z [v]. q

q

(12.53)

The duality relations are given in the cleanest form by Thm. 12.3.3 which can be translated into the following.

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Theorem 12.5.3. Let ` ∈ N and αj , βj ∈ N for all j = 1, . . . , `. For s = (α1 , 0β1 −1 , α2 , 0β2 −1 , . . . , α` , 0β` −1 ), we define its †-dual by Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 06/12/16. For personal use only.

s† = (β` , 0α` −1 , β`−1 , 0α`−1 −1 , . . . , β1 , 0α1 −1 ). Then we have †

ζqs [s] = ζqs [s† ]. By numerical computation with the help of some computer algebra system one can show that for all of the type ˜I, II and III q-MZVs duality relations are necessary to generate some Q-linear relations among q-MZVs that are missed by the DBSFs, at least when the weight is large enough. However, the combination of all the DBSFs and dualities are often not exhaustive yet. Sometimes, this difficulty can be overcome by increasing the weight and depth. But this seems to fail in some other cases, for example, for type ˜I q-MZVs of weight four. We can improve the above situation by considering the more general type G values. The advantage is that we have the new P-R relations which provide a lot of relations between type G q-MZVs, much more than the DBSFs and the duality combined in quantity (but may not provide new and linearly independent ones). The disadvantage is that there are too many type G values so that even when the weight is five the current computer power is too week to produce all the necessary relations. However, by using P-R relations all type G values can be converted to Q-linear combinations of type II values which can be handled by computers much easier. 12.6 Application to Okounkov’s Conjecture Using Cor. 12.2.15 we may regard Okounkov’s q-MZVs as Q-linear combinations of the q-MZVs ztq [s] for suitable auxiliary variable t. Further by using the P-R relations we may further reduce this to type II q-MZVs where we don’t need the letter π. Table 12.1: Dimension of type O q-MZVs, proved rigorously for w ≤ 6 and O numerically for w ≤ 12. All Re = DSO ≤w ∪ DU≤w . w ](W)O 6w dim ZO 6w dim All

2 1 1 0

3 2 2 0

4 4 4 0

5 7 7 0

6 12 11 1

7 20 18 2

8 33 27 6

9 54 42 12

10 88 63 25

11 143 95 48

12 232 142 90

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Applying the same idea as above it is possible to verify part (ii) of the following Okounkov’s Conjecture, at least when the weight is small. Conjecture 12.6.1. Let qZO be the Q-algebra generated by zO q [s], where s ∈ (N≥2 )d and d ≥ 1. Then (i) The algebra qZO is spanned by zO q [s] with 2 ≤ si ≤ 5, Z/2-graded by weight, and stable under the operator D that increases the weight by 2. O (ii) Let qZO ≤w be the Q-vector space generated by zq [s], |s| ≤ w. Then ∞ X w=0

tw dim qZO ≤w =

1 1 − 1−t−t2 +t6 +t8 −t13 1−t

= t2 +2t3 +4t4 +7t5 +11t6 +18t7 +27t8 +42t9 +63t10 +95t11 +142t12 +O(t13 ). For example, one can verify all of the following Q-linearly independent relations in the lower weight cases up to q 100 . One can also rigorously prove the first identity given by Eq. (12.54) involving only weight four and six values by using the relations we have found for type II q-MZVs (z = zO ): 4z[6] = z[2, 2] + 12z[3, 3] − 6z[4, 2],

(12.54)

4z[7] = z[2, 3] + z[3, 2] + 8z[3, 4] + 6z[4, 3] − 4z[5, 2], z[8] = z[2, 4] − z[6] + 2z[3, 3] + 6z[4, 4],

9z[8] = z[6] − 6z[3, 3] + 3z[4, 2] + 20z[3, 5] + 16z[5, 3] − 10z[6, 2], z[8] = 2z[2, 6] − z[6] + 2z[3, 3] + 4z[3, 5] − 16z[5, 3]

− 6z[2, 3, 3] + 3z[2, 4, 2] − 6z[3, 2, 3] − 3z[4, 2, 2],

4z[3, 6] = z[2, 5] + 4z[5, 2] + 3z[3, 4] + 6z[4, 5] + 8z[5, 4] + 2z[7, 2], 8z[9] = z[3, 4] − 5z[2, 5] − 8z[5, 2] − 30z[4, 5] − 2z[4, 3] − 36z[5, 4] − 10z[6, 3],

6z[4, 2] = 10z[6] + 42z[8] − 60z[2, 6] − 12z[3, 3] − 120z[3, 5] + 312z[5, 3]

− 15z[2, 2, 2] + 180z[2, 3, 3] − 90z[2, 4, 2] + 180z[3, 2, 3] + 60z[3, 3, 2],

72z[9] = 62z[5, 2] + 40z[2, 5] − 4z[3, 4] + 40z[3, 6] − 2z[4, 3] + 240z[4, 5] + 264z[5, 4] − 5z[2, 2, 3] − 60z[3, 3, 3] − 30z[4, 2, 3],

16z[9] = 2z[3, 4] − 10z[2, 5] − 12z[2, 7] − 8z[5, 2] − 60z[4, 5] − 24z[5, 4]

+ 4z[2, 3, 2] + 4z[3, 2, 2] + 3z[2, 2, 3] + 24z[2, 3, 4] + 18z[2, 4, 3] + 12z[3, 3, 3] − 12z[2, 5, 2] + 24z[3, 2, 4] + 6z[4, 3, 2],

64z[9] = 40z[2, 5] + 20z[2, 7] − 8z[3, 4] + 44z[5, 2] + 240z[4, 5] + 168z[5, 4]

+ 20z[3, 6]−4z[4, 3]−5z[2, 3, 2]−5z[2, 2, 3]−40z[2, 3, 4]−30z[2, 4, 3]

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+ 20z[2, 5, 2] − 5z[3, 2, 2] − 40z[3, 2, 4] − 100z[3, 3, 3] + 10z[3, 4, 2],

56z[9] = 30z[2, 5] + 20z[2, 7] + 26z[5, 2] − z[3, 4] + 180z[4, 5] + 112z[5, 4] Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 06/12/16. For personal use only.

+40z[3, 6]−6z[4, 3]−5z[2, 2, 3]−5z[2, 3, 2]−5z[3, 2, 2]−40z[2, 3, 4] + 20z[5, 2, 2] − 40z[3, 2, 4] − 30z[2, 4, 3] + 20z[2, 5, 2] − 140z[3, 3, 3].

Therefore, Conjecture 12.6.1 is proved rigorously up to weight 6 (inclusive), and verified numerically up to weight 12 (inclusive). The list of relations for weight 10 to 12 is too long to be presented here. 12.7 A q-Analog of Drinfeld Associator We have seen that the MZVs can be regarded as the coefficients of the Drinfeld associator related to the KZ equation given by Eq. (4.4) whose q-analog has the following form:   π πy 1 G(t) − G(qt) = + G(t). (12.55) t 1−q t 1−t

Note that

lim

q→1

1 G(t) − G(qt) = G0 (t). t 1−q

Theorem 12.7.1. The solution to Eq. (12.55) is given by X ∗ G(t) = ZIII q [w; t]w ∈ tQ[[t, q]]hhXπ ii.

(12.56)

∗ w∈Xπ

sd |s| III s1 where ZIII q [π y . . . π y; t] = (1 − q) zq [s; t] as defined in Eq. (12.2).

Proof. Let’s consider zIII q [s; qt]. There are two cases. If s1 = 1 then by k1 k1 k1 writing q t = t − (1 − q k1 )tk1 we have X q k1 tk1 zIII q [s; qt] = (1 − q k1 )(1 − q k1 )s2 · · · (1 − q kd )sd k1 >···>kd >0

= zIII q [s; t] − = zIII q [s; t] −

X

X

k2 >···>kd >0 k1 >k2

t 1−t

tk1 (1 − q k2 )s2 · · · (1 − q kd )sd

X k2 >···>kd >0

(1 −

q k2 )s2

tk2 · · · (1 − q kd )sd

t III z [s2 , . . . , sd ; t]. 1−t q Similar but simpler computation shows that if s1 > 1 then = zIII q [s; t] −

III III zIII q [s; qt] = zq [s; t] − zq [s1 − 1, s2 , . . . , sd ; t].

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Therefore, 

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t G(qt) = G(t) − (1 − q) π + πy 1−t

 G(t).

This is obviously the same as Eq. (12.55). The proof is now complete. Definition 12.7.2. We define the q-analog of the Drinfeld associator as X ∗ zIII Φq := (12.57) q [w; q]w ∈ tQ[[t, q]]hhXπ ii. ∗ w∈Xπ

12.8 q-MZVs by Bachmann and K¨ uhn 12.8.1

Mono-Brackets

It is well known that the Eisenstein series of weight s ≥ 4 Gs (τ ) :=

1 2

X (m,n)∈Z2 \{(0,0)}

∞ 1 (2πi)s X = ζ(s) + σs−1 (n)q n , (m + nτ )s (s − 1)! n=1

(12.58) P where τ ∈ H (the upper half-plane), q = e2πiτ , and σk (n) = d|n dk is the divisor sum function. To define the multiple Eisenstein series, we first generalize σk (n) to the multiple divisor sums as follows: X σr1 ,...,rl (n) = v1r1 . . . vlrl . (12.59) u1 v1 +···+ul vl =n u1 >···>ul ≥1

Definition 12.8.1. For any s1 , . . . , sl ∈ N, the generating function for the multiple divisor sum σs1 −1,...,sl −1 is defined by the formal power series [s1 , . . . , sl ] :=

X 1 σs1 −1,...,sl −1 (n)q n ∈ Q[[q]], (s1 − 1)! . . . (sl − 1)! n≥1

which is called a mono-bracket. Here we call |s| the weight and l the depth. For convenience, we set [∅] = 1. It is straightforward to see that [s1 , . . . , sl ] =

1 (s1 − 1)! · · · (sl − 1)!

X m1 ,...,ml ≥1 d1 ,...,dl ≥1

d1s1 −1 · · · dsl l −1

× q (m1 +···+ml )d1 +(m2 +···+ml )d2 +···+ml dl .

(12.60)

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By the Prop. 12.8.3, we will see that the mono-bracket [s] also provides a qanalog of MZV ζ(s). First, we define the sth modified Eulerian polynomial by  s ∞ X Ps (z) d z = z = ks z k (12.61) (1 − z)s+1 dz 1 − z k=1

where Ps (z) = z P˜s (z). We need to find its special value Ps (1) = P˜s (1) even though we do not need the full strength of the following lemma. Lemma 12.8.2. For any s ∈ N, we have P˜s (z) =

s−1 X

Es,n z n ,

n=0

where the Eulerian numbers Es,n are defined by   n X i s+1 (−1) (n + 1 − i)s . Es,n = i i=0 Moreover, Ps (1) = P˜s (1) = s!. Ps−1 Proof. Define Ps (z) = n=0 Es,n z n . Then P1 (z) = 1. We will show that P˜1 (z) = Ps (z). By the definition of Eq. (12.61) we see that P˜0 (z) = P˜1 (z) = 1. Now it is not hard to verify that Ps (z) satisfy the recurrence Ps+1 (z) = Ps (z)(1 + sz) + z(1 − z)Ps0 (z),

the same as that of P˜s (z), yielding P˜s (z) = Ps (z). This recurrence further implies that P˜s (1) = s!, as desired. Proposition 12.8.3. For any s ∈ Nl with s1 ≥ 2, we have X 1 lim (1 − q)|s| [s] = ζ(s) := . s1 − n1 · · · nsl l q→1 n1 >···>nl ≥1

Proof. Clear by Lemma 12.8.2. Definition 12.8.4. We define a normalized polylog by Lns (z) :=

Li1−s (z) , Γ(s)

where for z ∈ C, |z| < 1 the polylog Lis (z) is extended to s ∈ C: X zn Lis (z) = . ns n≥1

(12.62)

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The normalized polylog extends to an entire function in s and to a holomorphic function in z where |z| < 1. Proposition 12.8.5. For q ∈ C with |q| < 1 and for all s1 , . . . , sl ∈ N, we can write the mono-bracket as l X Y Psj −1 (q nj ) 1 (s1 − 1)! . . . (sl − 1)! n >···>n >0 j=1 (1 − q nj )sj 1 l X n1 = Lns1 (q ) . . . Lnsl (q nl )

[s1 , . . . , sl ] =

n1 >···>nl ≥1

where Pk (t) is the kth modified Eulerian polynomial. Proof. By the definition of Eq. (12.61), we have l Y Psj −1 (q nj ) = (1 − q nj )sj >···>n >0 j=1

X

n1

l

=

X

l X Y

s −1 vj nj

vj j

q

n1 >···>nl >0 j=1 vj ≥1

X

σs1 −1,...,sl −1 (n)q n .

n≥1

This yields the first equality in the proposition. The second follows from Definition 12.8.4. Corollary 12.8.6. If s1 ≥ 2 then

lim (1 − q)|s| [s] = ζ(s).

q→1−

Proof. This follows from Ps (1) = s! by Lemma 12.8.2 immediately. Definition 12.8.7. Let MB to be the n Q-vector space generated by [∅] o = 1 ∈ Q[[q]] and all the mono-brackets [s1 , . . . , sl ] : l ∈ N, s1 , . . . , sl ∈ N . In order to study the product structure of the mono-brackets we need the following lemma. Lemma 12.8.8. For a, b ∈ N, we have Lna (z) · Lnb (z) =

a X j=1

λja,b Lnj (z) +

b X

λjb,a Lnj (z) + Lna+b (z),

j=1

where λja,b = (−1)b−1



 a+b−j−1 Ba+b−j ∈ Q. (a + b − j)! a−j

(12.63)

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Proof. Set L(X) :=

X

Lnk (z)X k−1 =

k≥1 n≥1

k≥1

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X X nk−1 z n X eX z X k−1 = enX z n = . (k − 1)! 1 − eX z n≥1

Then 1 1 L(X) · L(Y ) = X−Y L(X) + Y −X L(Y ) e −1 e −1  L(X) − L(Y ) X Bn  (X − Y )n−1 L(X) + (Y − X)n−1 L(Y ) + = . n! X −Y n≥1

We can prove the lemma by comparing the coefficient of X a−1 Y b−1 . It is straightforward to see that the product of two mono-brackets can be found by using Prop. 12.8.5 and Lemma 12.8.8. Furthermore, similar to MZVs, mono-brackets together with their products can be formalized by using some algebra of words. Definition 12.8.9. Let alphabet AMB := {zj : j ∈ N}, A∗MB the set of words over AMB , and QhAMB i the noncommutative polynomial algebra over Q generated by A∗MB . Define  : A2MB → QhAMB i by za  zb = za+b +

a X

λja,b zj +

j=1

b X j=1

λjb,a zj ,

∀a, b ∈ N.

For all x, y ∈ AMB and u, v, w ∈ A∗MB , we define a multiplication on QhAMB i recursively by 1 w = w 1 = w and xu yv := x(u yv) + y(xu v) + (x  y)(u v). We define the weight of zs1 . . . zsl by s1 + · · · + sl and the depth by l. Proposition 12.8.10. The algebra (QhA∗MB i, ) is a commutative and associative Q-algebra. Proof. The proof is routine by using induction on the sum of depths of the words involved. For example, see the proof of [296, Thm. 2.1]. Proposition 12.8.11. Define a Q-linear map [ ] : (QhA∗MB i, ) → (MB, ·) so that [zs1 . . . zsl ] := [s1 , . . . , sl ]. Then we have [u v] = [u] · [v], Thus [ ] is a Q-algebra homomorphism.

∀u, v ∈ A∗MB .

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Proof. This follows by the same argument as in the MZV case by Lemma 12.8.8 and Prop. 12.8.5. Example 12.8.12. For any r, s, t ∈ N, we have [s] · [t] = [s, t] + [t, s] + [s + t] +

s X

λjs,t [j] +

j=1

t X

λjt,s [j]

j=1

and [r] · [s, t] = [r, s, t] + [s, r, t] + [s, t, r] + [s, r + t] + [r + s, t] +

r X

λjr,t [s, j] +

j=1

t X

λjt,r [s, j] +

j=1

r X

λjr,s [j, t] +

j=1

s X

λjs,r [j, t].

j=1

Considering the analogy between the MZVs and the mono-brackets we see that a missing link is the shuffle product structure. We will turn to this when we study the bi-brackets which are generalizations of mono-brackets. We know an admissible MZV ζ(s1 , . . . , sl ) must have s1 ≥ 2 so we define qMZ to be the subset of MB generated by the mono-brackets [s1 , . . . , sl ] with s1 ≥ 2. We expect that mono-brackets [1, . . . ] should behave quite differently from those in qMZ which is reflected in the next result. Proposition 12.8.13. The mono-bracket [1] is transcendental over qMZ. Proof. By Cor. 12.8.6 we see that if s1 ≥ 2 then near q = 1 we have [s] ∼ ζ(s)(1 − q)−|s| . We now claim that [1]  − Indeed, by the definition [1] =

∞ X

σ0 (n)q n =

n=1

log(1 − q) . 1−q ∞ X

q mn =

m,n=1

(12.64)

∞ X

qn . 1 − qn n=1

Thus [n] < n implies that ∞ ∞ 1 X qn log(1 − q) 1 X qn > =− . [1] = 1 − q n=1 [n] 1 − q n=1 n 1−q

(12.65)

On the other hand, define the sets SN for N ≥ 1 by the following. If N = b1 + q + · · · + q n c then n ∈ SN . With this notation, [1] ≤

1 1−q

b1/(1−q)c

X N =1

1 X n q . N n∈SN

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Let N∗ = min{n : n ∈ SN } and N ∗ = max{n : n ∈ SN }. Then

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∗

b1 + q + · · · + q N∗ c = b1 + q + · · · + q N∗ + q N∗ +1 + · · · + q N c. P ∗ Thus q N∗ +1 + · · · + q N < 1. Therefore n∈SN q n = q N∗ + q N∗ +1 + · · · + ∗ q N < q N∗ + 1 < 2 and !
1 Z 1−q b1/(1−q)c X 2 1 − log(1 − q) 2 1 2 dx [1] < < 1+ = . 1−q N 1−q x 1−q 1 N =1

Together with Eq. (12.65) this yields Eq. (12.64) which implies that [1] is transcendental over qMZ. Proposition 12.8.14. MB can be regarded as a polynomial ring over qMZ in the variable [1], i.e., MB = qMZ[[1]]. Proof. Let f = [{1}n , s1 , . . . , sl−n ], with s1 > 1. We now prove the proposition by induction on n. The case n = 0 is trivial so we assume n ≥ 1. But by the stuffle product given by Prop. 12.8.11 and the induction [1] · [{1}n−1 , s1 , . . . , sl−n ] − nf ∈ qMZ[[1]]. So f ∈ qMZ[[1]]. This finishes the proof of the proposition. 12.8.2

A Derivation

d In this subsection we turn to the operator D := q dq on MB and show it is a derivation and can provide some useful information of MB. For this purpose, we need to consider some finer structures of MB.

Definition 12.8.15. Let FilW • denote the increasing weight filtration on MB and FilD • the increasing depth filtration, namely, D E r FilW , k (MB) := [s] : s ∈ N , r ≥ 1, |s| ≤ k Q D E r FilD . l (MB) := [s] : s ∈ N , r ≤ l Q

If we consider both the depth and weight filtrations at the same time we D write FilW,D := FilW k Fill . k,l d is a derivation on MB, and Theorem 12.8.16. The operator D = q dq   W,D W,D D Filk,l (MB) ⊆ Filk+2,l+1 (MB).

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More precisely, let ⊕ be componentwise addition and ej = [. . . , 0, 1, 0, . . . ] where 1 only occurs at the jth component. Then for all s = [s1 , . . . , sl ] ∈ MB we have (setting sl+1 = 1 and ˜s = [s1 , . . . , sl+1 ]) D[s1 , . . . , sl ] = [2] · [s1 , . . . , sl ] − − −

j−1 l X X j=1 k=1 l X j=1

l+1 X

sj (˜s + ej ) +

j=1

(l − k + 1)sk (s + ek )

k=1

X

sk

l X

([s1 , . . . , sj−1 , α, β, sj+1 , . . . , sl ] + ek )

α+β=sj +1

(α − 1)

X

[s1 , . . . , sj−1 , α, β, sj+1 , . . . , sl ].

α+β=sj +2

Proof. For any fixed l ∈ N, define the generating series X

T (X1 , . . . , Xl ) :=

[s1 , . . . , sl ]

s −1

Xj j

j=1

s1 ,...,sl ≥1 l Y

X

=

l Y

n1 ,...,nl ≥1

enj Xj q n1 +···+nj . 1 − q n1 +···+nj j=1

Then we have T (X) · T (Y1 , . . . , Yl ) X = emX+n1 Y1 +···+nl Yl m,n1 ,...,nl ≥1

= T (X + Y1 , . . . , X + Yl , X) +

q n1 q n1 +···+nl qm . . . m n 1−q 1−q 1 1 − q n1 +···+nl

l X

T (X + Y1 , . . . , X + Yj , Yj , . . . , Yl )

j=1

+ Rl −

l X

T (X + Y1 , . . . , X + Yj , Yj+1 , . . . , Yl ),

(12.66)

j=1

where the last line comes from those terms when m = n1 + · · · + nj for some j so that   l l n1 +···+ni Pj Pl X X Y q   Rl = e k=1 nk (X+Yk )+ k=j+1 nk Yk n1 +···+ni )δi,j +1 (1 + q j=1 i=1 n1 ,...,nl ≥1

since 

qn 1 − qn

2 =

qn qn − . n 2 (1 − q ) 1 − qn

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The first line of Eq. (12.66) comes from the terms whose indices satisfy m > n1 + · · · + nl (put j = l + 1) or n1 + · · · + nj−1 < m < n1 + · · · + nj for some j. Setting m = n1 + · · · + nj−1 + m0 and nj = m0 + n0j for these terms we see the sum is now over m0 , n1 , . . . , n0j , . . . , nl which then gives T (X + Y1 , . . . , X + Yj , Yj
, . . . , Yl ). d Define D(f ) := dX f X=0 . Then D(Rl ) =

l X

X

(n1 + · · · + nj )e

Pl

k=1

nk Yk

j=1 n1 ,...,nl ≥1

l Y

q n1 +···+ni , (1 + q n1 +···+ni )δi,j +1 i=1

and this is exactly D T (Y1 , . . . , Yl ) which can be seen by induction on l and the Leibniz rule. The explicit expression of D is a straightforward computation and is thus left to the interested reader. This finishes the proof of the theorem. Corollary 12.8.17. The space qMZ is closed under D. Proof. These follow from straightforward computation. Example 12.8.18. We know that the Eulerian polynomials P1 (t) = t and P2 (t) = t(t + 1). Therefore we have t2 P2 (t) 1 P1 (t) = − . 3 3 (1 − t) 2(1 − t) 2 (1 − t)2

This quickly leads to

1 zIq [3] = [3] − [2] = [2, 1] − [2] + D[1] 2 using the first identity in Exercise 12.20. A surprising property of the derivation D is that it can provide many Q-linear relations between ordinary MZVs. To see this we define for any κ ≥ 1 the following spaces nX o an q n ∈ R[[q]] : an = O(nκ−1 ) , Fκ := n≥1

:=

nX

F 0 , o an q n ∈ R[[q]] : ∃ ε > 0 with an = O(nκ−1−ε ) .

Lemma 12.8.19. Let κ > 1. Then (i) All of F···>kd

d Y sgn(sj )kj q kj tj [kj ]|sj | ≥1 j=1

and ζq?,t [s] :=

X k1 ≥···≥kd

d Y sgn(sj )kj q kj tj . [kj ]|sj | ≥1 j=1

As we will only use ζq?,1 [s] for the star-type we will simply write ζ ? [s] = ζq?,1 [s] in the rest of this section.

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12.9.1

Mollified Companion of q-MZVs and q-MZ?Vs

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Adopting the notation from Chap. 9, we define the mollified companion of ζq1 [s] associated with the admissible triple of mollifiers (s; t; r) to be Z[s; t; r] := Z[s1 , . . . , sm ; t1 , . . . , tm ; r1 , . . . , rm ] m X Y sgn(sj )kj q tj kj +Q(rj ,kj ) (1 + q kj ) . = [kj ]|sj | k >···>k ≥1 j=1 1

m

?

If m = 0, we put ζ [∅] = Z[∅; ∅; ∅] = 1. Throughout this chapter the triples of mollifiers (s; t; r) are chosen in such a way that the above multiple series always converge. Although our primary goal is to prove q-MZ? V identities, in this chapter we will always work with the binomial identities for the q-MHS first. In order to obtain the corresponding q-MZ? V identities we need the next result. Lemma 12.9.1. Let 0 < q < 1, c, c1 , c2 ∈ R, c > 0, and let Rk be a sequence of real numbers satisfying |Rk | < k c1 q c2 k for all k = 1, 2, . . .. Then n ! n X ck2 k lim q 1 − n+k  Rk = 0. n→∞

k

k=1

Proof. For 1 ≤ k < n/2 and n sufficiently large, n (q; q)2n k 1 − n+k  =1− (q; q)n−k (q; q)n+k k =1−

(1 − q n−k+1 )(1 − q n−k+2 ) · · · (1 − q n ) = O(q c3 n ), (1 − q n+1 )(1 − q n+2 ) · · · (1 − q n+k )

where c3 is some positive constant independent of n. Therefore, we have n ! X n/2 X n/2 ck2 2 k < O(q c3 n ) q 1 − R q ck |Rk |   k n+k k=1 k k=1 < O(q

c3 n

)

n/2 X

k c2 q ck

2

+c1 k

.

(12.74)

k=1

On the other side, for n/2 ≤ k ≤ n, we can apply the trivial inequality n (1 − q n−k+1 )(1 − q n−k+2 ) · · · (1 − q n ) k 1. Suppose there is a triple of mollifiers ≈ ≈ e1 , . . . , λ em ; λ 1 , . . . , λ m ) satisfying λ = (λ1 , . . . , λm ; λ ≈

≈

λ 1
· · ·
λ j ∈ {1, 2}

∀j ≥ 1,

such that there is an expansion of the form ζ ? [s] = δ(s)Z] [λ]. Then for any integers c ≥ 1, we have ζ ? [c, s] =δ(s)Z] [Πc (λ)], where Πc (λ) are defined the same way as in Thm. 9.3.2. Using the same notation as on page 290 we get the following general result concerning the q-MZ? Vs. Theorem 12.9.4. Let s = (s1 , . . . , s` ) ∈ N` . If s1 > 1 then we have  1, if s` = 1; ≈ ? ] e 1 (s); m1 (s)], where δ(s) = ζ [s] = δ(s)Z [m1 (s); m −1, if s` ≥ 2. By letting q → 1 in Thm. 12.9.4 we can finally prove Thm. 10.8.1 which gives the corresponding general rule for the classical MZ? V in Chap. 10. Of course, to guarantee convergence we need to restrict a ≥ 1 there.

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12.9.2

Applications to MZVs and MZ?Vs

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By applying Lemma 12.9.1 to Cor. 9.4.3 we immediately get Corollary 12.9.5. With the same notation as in Cor. 9.4.3, we have ζ ? [s] = −Z] [Π(s)]. In particular, if c1 = c2 = . . . = c` = 3, then ζ ? [s] =

X

X

k1

(−1)k1 −1 q ( 2 )

(p,e p) k1 >···>km ≥1

m Y q pej kj (1 + q kj ) , [kj ]pj j=1

  ` ` e )  2a1 + 2, Cat{2aj + 3}, 2a`+1 + 1; Cat{aj + 1}, a`+1 . where (p; p j=2

j=1

The following result is a corollary of Exercise 9.3 and 9.4. Corollary 12.9.6. Let s = (s1 , . . . , sd ) ∈ Nd and s1 > 1. With the same notation as in Exercise 9.3 and 9.4, we have  1, if sd = 1; ? ] ζ [s] = δ(s)Z [Π(s)], where δ(s) = −1, if sd ≥ 2. For example, taking ` = 1 and c1 = 3, we get ζ ? [{2}b , 3, {2}a , 1] = Z] [2b + 2, 2a + 2; b + 1, a + 1; 1, 1],

ζ ? [{2}a0 , 1, {2}b , 3, {2}a1 , 1] ?

= Z] [2a0 + 1, 2b + 2, 2a1 + 2; a0 + 1, b + 1, a1 + 1; 2, −1, 1], b

a1

(12.77)

a2

ζ [{2} , 3, {2} , 1, {2} ]

= −Z] [2b + 2, 2a1 + 2, 2a2 ; b + 1, a1 + 1, a2 ; 1, 1, −1].

We can also obtain the following as the q-analog of Eq. (10.32). Corollary 12.9.7. Let a, b be two nonnegative integers. Then ζ ? [{2}a , 3, {2}b , 1] + ζ ? [{2}b , 3, {2}a , 1]

= ζ ? [{2}a+1 ]ζ ? [{2}b+1 ] + (1 − q)Z[2a + 2b + 3; a + b + 2; 2].

Proof. Taking n → ∞ and c = 2 in Eq. (9.15), we get, by Lemma 12.9.1, ?

a

ζ [{2} ] = Z[2a; a; 1] =

∞ X

(−1)

k=1

k−1 q

(k2)+ak (1 + q k ) [k]2a

.

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Thus ζ ? [{2}a , 3, {2}b , 1] + ζ ? [{2}b , 3, {2}a , 1] − ζ ? [{2}a+1 ]ζ ? [{2}b+1 ]
∞ X q (a+b+2)k+Q(2,k) 2(1 + q k ) − (1 + q k )2 = [k]2a+2b+4 k=1

=

∞ X q (a+b+2)k+Q(2,k) (1 + q k )(1 − q k )

[k]2a+2b+4

k=1

= (1 − q)Z[2a + 2b + 3; a + b + 2; 2] as desired. As a non-trivial example of Thm. 12.9.3 we may attach a string of type (2a , 1) to the front of the already treated type ({2}b , 1, {2}c , 3, {2}d , 1) given by Eq. (12.77) and get the following q-MZ? V identity: for any nonnegative integers a, b, c, d ζ ? [{2}a , 1, {2}b , 1, {2}c , 3, {2}d , 1] = Z] [2a + 1, 2b + 1, 2c + 2, 2d + 2; a + 1, b + 1, c + 1, d + 1; 2, 0, −1, 1]. Taking q → 1 in the above, we discover the classical MZ? V identity: ζ ? ({2}a , 1, {2}b , 1, {2}c , 3, {2}d , 1) = ζ ] (2a + 1, 2b + 1, 2c + 2, 2d + 2). By applying Lemma 12.9.1 to Cor. 9.4.3 we immediately get the following result. Theorem 12.9.8. Suppose ` ∈ N0 . Let s =

with aj , cj ∈ N0 and cj ≥ 3 for all j ≥ 1. Then



 `  Cat {2}aj , cj , {2}a`+1 j=1

ζ ? [s] = −Z] [Π(s)],  `  where Π(s) = 2a1 + 2, {1}c1 −3 , Cat 2aj + 3, {1}cj −3 , 2a`+1 + 1; j=2  `  ` Cat aj + 1, {0}cj −3 , a`+1 ; 1, Cat{θ}cj −2 . j=1

Proof. Clear.

j=1

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12.10 Historical Notes There is a long history of hunting for q-analogs of number theoretical objects in general. Concerning the multiple zeta values, Kaneko et al. first studied in detail the analytic properties and some special values of the type Iqanalog of the Riemann zeta functions in [337] in early 2000, which was generalized to multiple variable setting by the author [615] (first version was posted on arXiv in April 2003). In [615], he considered both the analytical and algebraic properties of type Iq-analog of the multiple zeta functions. He then defined their stuffle and shuffle products using the iterated Jackson q-integrals although only very few DBSFs could be found as the shuffle is quite complicated. In 2004, Bradley independently studied type Iq-MZVs in [81] by concentrating on the generalization of classical identities of the MZVs to this q-analog setting, and later, Okuda and Takeyama also looked at some of the relations among this type of the q-MZVs in [456]. Before the study of type Iq-MZVs, slightly different definitions of the q-MZVs were already presented in [499, 643]. Additionally, it is not hard to see that Schlesinger’s version diverges when |q| < 1 but can converge if |q| > 1. In fact, for s = (s1 , . . . , sd ) ∈ Zd (0,...,0)

zq−1

|s| II [s] = (−1)|s| z(s) q [s] = (−1) zq [s].

So it suffices to consider type II first defined in [643] in order to understand Schlesinger’s q-MZVs. The analytic properties of type Iq-MZVs are treated in [615] which also includes the stuffle relation and the shifting principle. However, the general shuffle structure was not defined until Takeyama’s work in Ref. [530], from which our definition for ∗˜I is adopted. Unfortunately, his approach to the shuffle relations relies on some auxiliary multiple polylog functions and consequently it is very hard to see why these relations should hold. The situation looks much better with the appearance of the joint work [120] by Castillo Medina et al. who generalized Chen’s iterated integrals to Jackson’s iterated q-integrals to study type III q-MZVs by using Rota– Baxter algebra technique, which is modified and generalized in Sec. 12.2. Further study was carried out in [515] by Singer along this line. This was later modified to treat all the general q-MZVs in [632]. In 1958, influenced by K.-T. Chen’s systematic investigation the iterated integrals, Ree [481] first considered algebras in which the product is expressed in terms of shuffles. Later, G. Baxter [41] constructed algebras with an operator P satisfying Eq. (12.4) with λ = 1. In general, Eq. (12.4) was defined and popularized by Rota and his school. See [487–489], or

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the concise survey by Guo [257] and Chap. 3 of his book [258]. In [179] Ebrahimi-Fard and Guo considered MZVs and their generalizations from the point of view of Rota–Baxter algebras and derived some relations on MZVs from those in Rota–Baxter algebras. Linear operators satisfying Eq. (12.4) in the context of Lie algebras were introduced in the 1980s and were linked to the solutions of the classical Yang–Baxter equation (after C. Yang and R. Baxter). Along this line, Connes and Kreimer applied the Hopf algebra technique and developed a renormalization theory in perturbative quantum field theory (see [147,148]) which led to some similar works on renormalizations of the MZVs in [264] and their q-analogs in [618]. There are some further generalizations and some other approaches to the renormalization problem of the MZVs, for example, see [176, 180–183, 413, 414]. In [184], we showed that there is a group acting on the space of all renormalizations of MZVs transitively. Note that the Jackson q-integral was first defined in [315]. However, his difference operator in [316] is different from ours in this chapter. In [530], Takeyama also noticed the set of type Iq-MZVs needs to be enlarged to type ˜I q-MZVs which are a kind of “regularized” q-MZVs in the sense that one needs to consider some convergent versions of the q-MZVs when s1 = 1 by modifying the auxiliary variables of t. But for these type ˜I q-MZVs themselves, the DBSFs are insufficient to provide all the Q-linear relations and a certain “Resummation Identity” defined by Takeyama is required. In this book, we adopt the term “duality” due to its similarity to the duality relations of the ordinary MZVs. Concerning the individual results, Thm. 12.2.11 is parallel to [120, Prop. 9] and includes [530, Thm. 1] as a special type ˜I case. Proposition 12.2.18 is [120, Thm. 7] while Corollary 12.2.19 generalizes [530, Prop. 1]. The case for type ˜I of Prop. 12.2.20 was established by Takeyama as [530, Prop. 2.4]. Note that Thm. 12.2.21 generalizes [530, Thm. 2] but it does not contain [120, Thm. 7] since our word representation of type III q-MZVs in this book is different from that of [120]. Theorem 12.3.1 was called by Takeyama as the “Resummation Identity” [530, Thm. 4]. Turning to the more complicated q-MZVs, Okounkov stated Conjecture 12.6.1 in [454] after guessing that his version of the q-MZVs is related to some q-series invariants of the Hilbert schemes parameterizing 0dimensional subschemes of finite lengths on a nonsingular quasi-projective surface. This conjectured relation to Hilbert schemes has been proved by Qin and Yu in [476] when the surface is abelian or, in general, modulo lower weight terms. Our definition of Okounkov’s q-MZV here is equal to

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his original q-MZVs up to a suitable 2-power. Some special cases of part (ii) of Okounkov’s Conjecture 12.6.1 on the stability of the operator D was proved by Bachmann and K¨ uhn [27]. The double shuffle relations of the double Eisenstein series were first considered by Gangl et al. in [233]. The level two case was studied by Kaneko and Tasaka [334]. Yuan and the author extended this to arbitrary levels in [590]. The general multiple Eisenstein series first appeared in the joint works of Bachmann and K¨ uhn [24, 26] via their mono-brackets. The stuffle relations are obvious. However, it is much harder to define the shuffle product structure since no integral expressions for the multiple Eisenstein series are available. Bachmann and Tasaka partially developed such a theory by using the Hopf algebra structure of the (generic) iterated integrals introduced by Goncharov. We refer the interested reader to their original paper [28] for details. The multiple divisor functions used in the definition of the multiple Eisenstein series are generalized to arbitrary levels in [591], through which the MZVs are extended to Hurwitz-type values and their DBSFs are studied. To help to determine the DBSFs of the mono-brackets, Bachmann introduced the bi-brackets in [25]. Then Zudilin [644] constructed the word algebras that encode the DBSFs of the bi-brackets. In particular, his duality in Proposition 12.8.32 is exactly the partition duality of Bachmann. However, we still need to understand how to extend this to the mono-bracket, one way of which is to solve Conjecture 12.8.27 of Bachmann. All the results in Sec. 12.9 are consequences of the corresponding results of the q-MH? S obtained in Chap. 9. They are based on the joint work in [280], which generalizes [277], the author’s work [628], and the joint work [402] for the ordinary MHS, MZVs and MZ? Vs. In particular, the mollified companions essentially already appeared in [277] and [280]. The key is Lemma 12.9.1 obtained in [277] which generalizes a similar result in the classical non-q setting appearing first as [402, Lemma 4.2]. Finally, some other families of the MZV identities we encountered in Chap. 5 have been extended to the q-setting. Due to space limitation, we refer the interested reader to the Refs. [79–81, 119, 169, 277, 280, 396, 449, 456, 615]. Exercises 12.1. There are many interesting relations among type Iq-MZVs. The following is the q-analog of the cyclic sum formula (cf. Thm. 5.2.3). Let

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s = (s1 , . . . , sd ) ∈ Nd , s1 ≥ 2. Then

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d X

ζqI [si + 1, si+1 , . . . , sd , s1 , . . . , si−1 ]

i=1

=

d sX i −2 X i=1 j=0

ζqI [si − j, si+1 , . . . , sd , s1 , . . . , si−1 , j + 1].

12.2. Show that the stuffle products ∗τ is well defined for τ = ˜I, II or III. 12.3. The following result gives the q-analog of Ohno’s Relations (cf. Thm. 5.5.9). Let s = (s1 , . . . , sd ) ∈ Nd , s1 ≥ 2 and its dual s∗ = (s01 , . . . , s0d0 ) (see Eq. (5.1)). Then for any nonnegative integer l X X ζqI [s1 +e1 , . . . , sd +ed ] = ζqI [k10 +e01 , . . . , kd0 0 +e0d0 ]. e01 +···+e0d0 =l e0i ≥0

e1 +···+ed =l ei ≥0

12.4. Prove the following q-analog of Euler’s decomposition formula. For all a, b ∈ N, s−1 s−1−a X a + t − 1t − 1 X I I zIq [t + a, s − a − b] zq [s]zq [t] = t − 1 b a=0 b=0

+

t−1 t−1−a X  X a=0

b=0

  a+s−1 s−1 I zq [s + a, t − a − b] s−1 b

min(s,t)

−

X (s + t − j − 1)! 1 1 · ϕq [s + t − j], (s − j)! (t − j)! (j − 1)! (1 − q)s+t−j j=1

where ϕq is defined by ϕq [n] =

∞ X

(k − 1)

k=1

q (n−1)k . [k]n

(12.78)

12.5. Verify Eqs. (12.27), (12.28) and (12.29) by direct computation. 12.6. Equation (12.12) is commonly considered to be the q-analog for the classical integration by parts rule:

R(f )R(g) = R f R(g) + R R(f )g , (12.79) which can be seen as dual to Leibniz rule for usual derivations. R t For any fixed constant a, show that Eq. (12.79) holds if we set R(f )(t) = a f (x) dx.

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12.7. Prove that limq→1 Jq [f ](t) =

Rt 0

f (x) dx.

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12.8. Prove Prop. 12.2.2 by direct computation. 12.9. Prove Cor. 12.2.17. 12.10. Show that the shifting operator defined by Eq. (12.23) is injective. 12.11. Prove the following q-analog of Euler’s decomposition formula. For all a, b ∈ N, III zIII q [a]zq [b]

=

a−1 X a−1−l X l=0

+ −

(−1)

k=0

k



l+b−1 b−1

b−1 X

min(a,b−1−l)

l=0

k=0

a X k=1

X

k

(−1)



  b III z [b + l, a − l − k] k q

  l + a − 1 a III z [a + l, b − l − k] a−1 k q

βa−k zIII q [a + b − k] +

a−1 X j=1

αa+b−1−j D zIII q [a + b − 1 − j],

d and where D := q dq

 k X (−1)a+b−j

 j−1 , αk = 1−j j − b, j − a, a + b − j − 1 j=b   j+b−1 a−j . βj = (−1) j, j + b − a, a − j − 1 12.12. Prove the commutativity of the q-shuffle product by showing that u v = v u for all u, v ∈ Ye ∗ using induction on the sum |u| + |v|.







12.13. Show that (A0G , ∗G ) is a commutative and associative algebra. 12.14. Define another type of the q-MZVs by 1 −1,s2 ,...,sd ) z(s [s1 , . . . , sd ], q

s1 ≥ 1, s2 , . . . , sd ≥ 0.

Define the shuffle and stuffle structures of these q-MZVs by mimicking the treatment of type Iq-MZVs. 12.15. Prove Prop. 12.8.26 by induction on the sum of the weights of the words involved. In fact, if we change the definition of bi-brackets by using the increasing order on the indices in Eq. (12.68) then this is equivalent to the definition in Eq. (12.69).

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12.16. Denote by Zτ≤w the Q-space generated by q-MZVs of type τ correτ sponding to the type τ -admissible words W≤w , DSτ≤w the space generated τ by the DBSFs, and DU≤w the space generated by the duality relations. Hence DUτ≤w \ DSτ≤w gives the duality relations that are not contained in DSτ≤w . Using your favorite computer algebra system to confirm the following data contained in Tables 12.2-12.5.

Table 12.2: Dimension of the q-MZVs of type ˜I. w ˜ ](W)I6w ˜ lower bound of dim ZI6w

2 4 3

3 12 7

4 33 14

5 88 27

6 232 50

7 609 91

dim DSI6w

1

4

17

56

171

497

0

0

1

2

3

6

˜

dim

˜ DUI6w

\

˜
DSI6w

Table 12.3: Dimension of the q-MZVs of type I. w ](W)I6w lower bound of dim ZI6w dim DSI6w
dim DUI6w \ DSI6w

2 1 1 0 0

3 3 2 1 0

4 7 4 3 0

5 15 7 8 0

6 31 11 20 0

7 63 18 45 0

8 127 27

Table 12.4: Dimension of the q-MZVs of type II. w ](W)II6w lower bound of dim ZII6w dim DSII6w
dim DUII6w \ DSII6w

1 1 1 0 0

2 5 3 1 1

3 19 12 5 2

4 69 30 28 8

5 251 73 124 35

6 923 173 536 127

9 255 42

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Table 12.5: Dimension of the q-MZVs of type III. w ](W)III 6w lower bound of dim ZIII 6w dim DSIII 6w
III dim DUIII 6w \ DS6w

1 1 1 0 0

2 5 4 1 0

3 19 12 5 1

4 69 30 28 1

5 251 73 124 5

6 923 173 536 4

12.17. For s1 , s2 with s1 + s2 > 2 and s = s1 + s2 − 2, show that     s s D[s] = [s1 ] · [s2 ] + [s + 1] s1 − 1 s s1 − 1     X a−1 a−1 − + [a, b]. s1 − 1 s2 − 1 a+b=s+2

12.18. Complete the proof of Prop. 12.8.11. 12.19. For s1 , s2 ∈ N, show that D[s1 , s2 ] = [2] · [s1 , s2 ] − s1 [s1 + 1, s2 , 1] − s2 [s1 , s2 + 1, 1] − [s1 , s2 , 2] + 2s1 [s1 + 1, s2 ] + s2 [s1 , s2 + 1] −

X

(a − 1)[a, b, s2 ]

a+b=s1 +2

! X

+

a+b=s2 +1

s1 [s1 + 1, a, b] +

X

(a − 1)[s1 , a, b] .

a+b=s2 +2

12.20. Show that 1 D[1] = [3] + [2] − [2, 1], 2 1 D[2] = [4] + 2[3] − [2] − 4[3, 1], 6 1 D[2] = 2[4] + [3] + [2] − 2[2, 2] − 2[3, 1], 6 3 1 D[1, 1] = [3, 1] + [2, 1] + [1, 2] + [1, 3] − 2[2, 1, 1] − [1, 2, 1], 2 2 1 3 D[1, 2] = − [1, 2] + 2[1, 3] + [1, 4] + [2, 2] + [3, 2] 6 2 − 4[1, 3, 1] − [2, 1, 2] − 2[2, 2, 1].

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12.21. The algebra qMZ view as a Q-subalgebra of MB is filtered by the weight whose graded w piece is denoted by grw qMZ. Let dw be its dimension over Q. Is the following true? ∞ X

dw tw =

w=0

1 − t2 + t4 . 1 − 2t2 − 2t3

Verify some lower weight cases. 12.22. For any real number r, we define [r]q := (q r − 1)/(q − 1). Then for all |x|, |y| < 1 we have ∞ X ∞ X

a=0 b=0

I b (−1)a+b [x]a+1 [x]b+1 q q ζ [a + 2, {1} ] = 1 −

Γq (1 + x)Γq (1 + y) Γq (1 + x + y)

where the q-analog of the Gamma function 1−x

Γq (x) := (1 − q)

∞ Y 1 − q i+1 . 1 − q i+x i=0

12.23. Use Exercise 12.22 to establish the following expression of the generating function of the height one q-MZVs: ∞ ∞ X X a=0 b=0

ua+1 v b+1 ζ I [a + 2, {1}b ] 

= 1 − exp 

∞ X 1 i=2

i

ui + v i − (u + v + (1 − q)uv)i

i X j=2

 (q − 1)i−j ζ I [j] .

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Chapter 13

Colored Multiple Zeta Values

In Chap. 2 we have studied mainly the analytic properties of the multiple polylog functions. In the following two chapters we turn to the algebraic side of the special values of these functions. 13.1 Definitions Recall from Definition 2.3.1 that for any positive integers s1 , . . . , sd , the multiple polylog of complex variables is defined as follows: X

Lis1 ,...,sd (z1 , . . . , zd ) :=

k1 >···>kd >0

z1k1 · · · zdkd , k1s1 · · · kdsd

(13.1)

where |z1 · · · zj | < 1 for j = 1, . . . , d. It is analytically continued to a multivalued meromorphic function on Cd in Sec. 2.4. √ Definition 13.1.1. Fix an N th root of unity µ = µN := exp(2π −1/N ). A colored multiple zeta value (CMZV) of level N is a number of the form LN (s1 , . . . , sd |i1 , . . . , id ) := Lis1 ,...,sd (µi1 , . . . , µid ).

(13.2)

We will always identify (i1 , . . . , id ) with (i1 , . . . , id ) (mod N ). The N th roots of unity are called the colors. As before, d is called the depth and s1 + · · · + sd the weight. It is easy to see from Eq. (13.1) that a CMZV converges if and only if (s1 , µi1 ) 6= (1, 1). Clearly, all CMZVs of level N are automatically of level kN for every positive integer k. For example, when i1 = · · · = id = 0 or N = 1 one gets the MZV ζ(s1 , . . . , sd ). When N = 2 one recovers socalled Euler sums which will be treated in section 14.2 and whose partial sums have been studied in Chap. 7. We also have the following integral 419

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expression by Eq. (2.8):

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LN (s1 , . . . , sd |i1 , . . . , id ) = where al =

Ql

j=1

Z 0

1



dt t

s1 −1

dt ··· a1 − t



dt t

sd −1

dt ad − t (13.3)

µ−ij for l = 1, . . . , d.

13.2 Dimension Upper Bound by Deligne and Goncharov Similar to the MZV case, standard conjectures in arithmetic algebraic geometry imply that all Q-linear relations among CMZVs can be derived from those among CMZVs of the same weight. Let CMZV(w, N ) be the Q-span of all the CMZVs of weight w and level N . Let d(w, N ) denote its dimension. In general, it is very difficult to determine d(w, N ) because any nontrivial lower bound would be equivalent to some nontrivial irrational/transcendental result which in turn would be a consequence of a variant of Grothendieck’s period conjecture. For example, one can √ √ show easily that CMZV(2, 4) = hlog2 2, π 2 , π −1 log 2, (K − 1) −1i, where P n 2 K = n≥0 (−1) /(2n + 1) is the Catalan’s constant. From the abovementioned Grothendieck’s conjecture we know d(2, 4) = 4 (see [161] or [240]) but we don’t have an unconditional proof of this equlity yet. Namely, √ √ we cannot prove that the four numbers log2 2, π 2 , π −1 log 2, (K − 1) −1 are linearly independent over Q. Consequently, a nontrivial lower bound of d(w, N ) is hard to come by. On the other hand, one may obtain upper bounds of d(w, N ) by finding as many Q-linear relations in CMZV(w, N ) as possible. As in the cases of the MZVs and the Euler sums the double shuffle relations will play important roles in revealing the relations among CMZVs. If such a relation is produced by multiplying two convergent CMZVs then it is called a finite double shuffle relation (finite DBSF) similar to the MZV case. In general, we need to regularize the divergent CMZVs to obtain the regularized double shuffle relations (regularized DBSFs). We shall recall this theory in Sec. 13.3.1. The proof of the following important result of Deligne and Goncharov is far beyond the scope of this book, so we omit it here. Theorem 13.2.1. For any weight w ≥ 1 and d(w, N ) ≤ D(w, N ) where D(w, N ) are defined by  2 3 −1 ∞  (1 − t − t ) , X 1+ D(w, N )tw = (1 − t − t2 )−1 ,  w=1 (1 − at + bt2 )−1 ,

level N ≥ 1, we have the formal power series if N = 1; if N = 2; if N ≥ 3,

(13.4)

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where a = a(N ) := ϕ(N )/2 + ν(N ), b = b(N ) := ν(N ) − 1, ϕ is Euler’s totient function and ν(N ) is the number of distinct prime factors of N .

Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 06/12/16. For personal use only.

If N > 2 then we have ∞ X w=1

D(w, N )tw = at + (a2 − b)t2 + (a3 − 2ab)t3 + (a4 − 3a2 b + b2 )t4 + · · ·

In particular, if p is a prime then for any positive integer n w  n−1 p (p − 1) . +1 D(w, pn ) = a(pn )w = 2

(13.5)

We will compare the bound D(w, N ) of Deligne and Goncharov to the bound obtained by the standard relations to be defined next. 13.3 Standard Relations It turns out that in higher level cases, the DBSFs are far from enough to produce all the Q-linear relations between CMZVs. In this section, we list some more relations that can be defined without too much difficulty. Definition 13.3.1. We call a Q-linear relation between CMZVs standard 1 if it can be produced by Q-linear combinations of the following four families of relations: the regularized DBSFs (Sec. 13.3.1), the regularized distribution relations (regularized DISTs) (Sec. 13.3.4), the weight one relations (Sec. 13.3.3), and the lifted relations from the above three types (Sec. 13.3.5). Otherwise, it is called a non-standard relation. It is commonly believed that all linear relations among the MZVs (i.e., level one CMZVs) are consequences of the regularized DBSFs. But when level N ≥ 2 we don’t know, even conjecturally, how to produce all linear relations among the CMZVs. Moreover, when N = 2 (i.e., for the Euler sums) we don’t know whether there are any inclusion relations among the regularized DBSFs, the regularized DISTs, the doubling relations and the generalized doubling relations (see Sec. 14.2.5). 13.3.1

Regularized Double Shuffle Relations

Similar to the cases of the MZVs, we first define some algebras which reflect the behavior of multiplications of CMZVs. 1 This term was suggested by P. Deligne in a letter to Goncharov and Racinet dated Feb. 25, 2008.

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Set

dt µi dt , bi = for i = 0, 1, . . . , N − 1. t 1 − µi t For every positive integer k ≥ 1, define the word of length k

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a=

zk,i := ak−1 bi .

Then it is straightforward to verify using Eq. (13.3) that if (s1 , µi1 ) 6= (1, 1) then Z 1 LN (s1 , . . . , sn |i1 , i2 , . . . , in ) = zs1 ,i1 zs2 ,i1 +i2 · · · zsn ,i1 +i2 +···+in . (13.6) 0

One can now define an algebra of words as follows (cf. Definition 7.2.1): Definition 13.3.2. The set of alphabet X consists of N + 1 letters a and bi for i ≡ 0, . . . , N − 1 (mod N ). The weight of a word w (i.e., a monomial in the letters in X denoted by |w|) is the number of letters contained in w, and its depth, denoted by dp(w), is the number of bi ’s contained in w. Define the Hoffman–Racinet algebra of level N , denoted by AN , to be the (weight) graded noncommutative polynomial Q-algebra generated by words over the alphabet X. Let A0N be the subalgebra of AN generated by words not beginning with b0 and not ending with a. The words in A0N are called admissible words. By Eq. (13.6) every CMZV can be expressed as an iterated integral over the closed interval [0, 1] of an admissible word w in A0N . This is denoted by Z 1 L(w) := w. (13.7) 0

We also extend L to AN by Q-linearity. We remark that the length lg(w) of w is equal to the weight of L(w). Therefore in general one has LN (s1 , . . . , sn |i1 , i2 , . . . , in ) = L(zs1 ,i1 zs2 ,i1 +i2 · · · zsn ,i1 +i2 +···+in ), (13.8)

L(zs1 ,i1 zs2 ,i2 · · · zsn ,in ) = LN (s1 , . . . , sn |i1 , i2 − i1 , . . . , in − in−1 ). (13.9)

For example, L3 (1, 2, 2|1, 0, 2) = L(z1,1 z2,1 z2,0 ). Therefore, we can apply Chen’s theory of the iterated integrals to compute the product of two CMZVs. For example, by Lemma 2.1.2 from Chap. 2, one has LN (1|1)LN (2, 3|1, 2) = L(z1,1 ) L(z2,1 z3,3 ) = L(b1 2

= L(b1 ab1 a b3 +

2ab21 a2 b3

2


(ab a b )) 1

2

3

2

+ (ab1 ) ab3 + ab1 a b1 b3 + ab1 a2 b3 b1 )

= L(z1,1 z2,1 z3,3 + 2z2,1 z1,1 z3,3 + z22,1 z2,3 + z2,1 z3,1 z1,3 + z2,1 z3,3 z1,1 ) = LN (1, 2, 3|1, 0, 2) + 2LN (2, 1, 3|1, 0, 2) + LN (2, 2, 2|1, 0, 2) + LN (2, 3, 1|1, 0, 2) + LN (2, 3, 1|1, 2, N − 2).

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Let AN,
be the algebra of AN where the multiplication is defined by the shuffle product . Denote the subalgebra A0N by A0N,
when one considers the shuffle product. Then one can easily prove Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 06/12/16. For personal use only.




Proposition 13.3.3. The map L : A0N,
−→ C is an algebra homomorphism. Proof. See Exercise 13.2. On the other hand, the CMZVs are known to satisfy the series stuffle relations. For example, LN (2|5)LN (3|4) = LN (2, 3|5, 4) + LN (3, 2|4, 5) + LN (5|9). To study such relations in general one has the following definition. Definition 13.3.4. Denote by A1N the subalgebra of AN which is generated by words zs,i with s ∈ N and i ≡ 0, . . . , N − 1 (mod N ). Equivalently, A1N is the subalgebra of AN generated by words not ending with a. For any word w = zs1 ,i1 zs2 ,i2 · · · zsn ,in ∈ A1N and positive integer j, one defines the exponent shifting operator τj by τj (w) = zs1 ,j+i1 zs2 ,j+i2 · · · zsn ,j+in . For convenience, on the empty word we adopt the convention that τj (1) = 1. We then define another multiplication ∗ on A1N by requiring that ∗ distribute over addition, that 1 ∗ w = w ∗ 1 = w for any word w, and that, for any words u, v,  

 zs,j u ∗ zt,k v = zs,j τj τ−j (u) ∗ zt,k v + zt,k τk zs,j u ∗ τ−k (v) 
 + zs+t,j+k τj+k τ−j (u) ∗ τ−k (v) .

(13.10)

This multiplication is called the stuffle product. If we denote by A1N,∗ the algebra (A1N , ∗) then it is not hard to prove the next proposition. Proposition 13.3.5. The polynomial algebra A1N,∗ is a commutative weight graded Q-algebra. Proof. See Exercise 13.3.

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Now we can define the subalgebra A0N,∗ similar to A0N,
by replacing the shuffle product by the stuffle product. Then by the induction on the lengths and using the series definition one can quickly check that for any w1 , w2 ∈ A0N,∗ L(w1 ) L(w2 ) = L(w1 ∗ w2 ). This implies the following result. Proposition 13.3.6. The map L : A0N,∗ −→ C is an algebra homomorphism. Proof. See Exercise 13.3. Definition 13.3.7. Let w be an integer such that w ≥ 2. For nontrivial words w1 , w2 ∈ A0N with |w1 | + |w2 | = w, we say that the equation L(w1


w

2

− w1 ∗ w2 ) = 0

(13.11)

provides a finite double shuffle relation (finite DBSF) of weight w. It is known that even at level one (i.e., the MZV case) these relations are not enough to provide all the relations among the MZVs. For example, the weight of the product of two MZVs is at least 4. So the finite DBSFs cannot imply the well-known identity ζ(2, 1) = ζ(3). However, it is believed that one can remedy this by considering regularized double shuffle relation (regularized DBSF) produced by the following mechanism. First, combining Prop. 13.3.3 and Prop. 13.3.6 we can easily prove the following algebraic result. Proposition 13.3.8. We have two algebra homomorphisms: L∗ : (A1N,∗ , ∗) −→ C[T ],

and

L
: (A1N,
,


) −→ C[T ]

which are uniquely determined by the properties that they both extend the evaluation map L : A0N −→ C by sending b0 = z1,1 to T .

Second, in order to establish the crucial relation between L∗ and L
one can adopt the machinery similar to that of the MZVs developed in section 3.2 as follows. For any (s|i) = (s1 , . . . , sn |i1 , . . . , in ) where ij are integers and sj are positive integers, let the image of the corresponding words in A1N under L∗ and L
be denoted by L∗ (s|i; T ) and L
(s|i; T ) respectively.

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Theorem 13.3.9. Define a C-linear map ρ : C[T ] → C[T ] by ! ∞ X (−1)n n Tu ζ(n)u eT u , |u| < 1. ρ(e ) = exp n n=2 Then for any index set (s|i) one has
L
(s|i; T ) = ρ L∗ (s|q(i); T ) ,

(13.12)

where q(i1 , . . . , in ) = (i1 , i2 − i1 , . . . , in − in−1 ). Thm. 13.3.9 is a generalization of Thm. 3.3.21 to the higher level CMZV cases, but the proof is much more complicated, which will be given in the next section using some advanced results from Lie algebra. Definition 13.3.10. Let w be a positive integer such that w ≥ 2. Let (s|i) be any index set with the weight of s equal to w. Then every weight w CMZV relation produced by Eq. (13.12) is called a regularized double shuffle relation (regularized DBSF) of weight w. This is obtained by formally setting T = 0 in Eq. (13.12). The above steps can be easily transformed to computer codes which have been used in some Maple programs (see Appendix D.1). For example, one gets by the stuffle product T LN (2|3) = L∗ (1|0; T ) L∗ (2|3; T ) = L∗ (z1,0 ∗ z2,3 )

= L∗ (1, 2|0, 3; T ) + L∗ (2, 1|3, 3; T ) + L∗ (3|3; T ),

while using shuffle product one has T LN (2|3) = L
(1|0; T ) L
(2|3; T ) = L
(z1,0


z

2,3 )

= L
(b0


ab ) 3

= L
(1, 2|0, 3; T ) + L
(2, 1|0, 3; T ) + L
(2, 1|3, 0; T ).

We thus obtains the following regularized DBSF if we compare the above two expressions using Thm. 13.3.9: LN (2, 1|3, 0) + LN (3|3) = LN (2, 1|3, N − 3) + LN (2, 1|0, 3). 13.3.2

Proof of Regularized Double Shuffle Relations

When N = 1 most results in this subsection have been proved in Chap. 3 already. But the proofs of these results for general levels N contained here are often different because more advanced tools are needed. Recall that we have defined the alphabet X = XN = {a, bj : j ∈ Z/N Z} √ and chosen µ = µN = exp(2π −1/N ). Let ΓN := {µj : j ∈ Z/N Z} be

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the set of N th roots of unity. For convenience, we also use the letters xξ , ξ ∈ ΓN ∪{0}, such that x0 = a and xµj = bj . Thus X = {xξ : ξ ∈ ΓN ∪{0}}. We now denote by Y the subset of X∗ of all the words yk,ξ = xk−1 xξ for 0 k ∈ N and ξ ∈ ΓN . Remark 13.3.11. Observe that yn,µj = zn,j used in the preceding subsection. It turns out that the notation yn,ξ adopted throughout this subsection has some theoretical advantage while it is much easier to use zn,j in computer aided symbolic computations which have been implicitly used in the examples of the preceding subsection. Let R be any Q-algebra. The subalgebra of RhXi generated by Y is denoted by RhYi and is free on Y. As a vector space, it is generated by the words over X not ending with x0 thus it could be identified with the quotient RhXi/RhXix0 . The corresponding projection is denoted by πY . The vector subspace RhXicv of RhXi generated by words over X that are not ending with x0 and not starting with x1 is also a graded algebra. It may be identified with the quotient RhXi/(x1 RhXi + RhXix0 ). We denote by πcv the corresponding projection. The algebra RhXicv is same as the subalgebra of RhYi generated by words over Y that are not starting with y1,1 = x1 . (Note that the second subscript 1 in y1,1 is an element in ΓN .) Thus it can be identified with the quotient RhYi/y1,1 RhYi, which is denoted by RhYicv . Again its corresponding projection is denoted by πcv . Similarly one defines RhhXiicv and RhhYiicv as the completion of RhXicv and RhYicv with respect to the weight grading, respectively. Remark 13.3.12. The algebras A, A0 and A1 in Sec. 13.3.1 are essentially equal to QhXi, QhXicv and QhYi, respectively. Now we take (ChXi, •, ∆) to be the graded bialgebra defined by Example B.3 in Appendix B and (ChYi, •, ∆∗ ) the graded bialgebra defined by Example B.4. In particular, all xξ are primitive elements in (ChXi, •, ∆). Similarly, ChhXii, ChhYii, ChhXiicv and ChhYiicv are all graded bialgebras (see Exercise 13.4). Recall that for any a1 , . . . , ak ∈ C and any path γ : [0, 1] → C, the iterated integral Iγ (a1 , . . . , ak ) is defined by Eq. (2.14). In particular, for ξ1 , . . . , ξd ∈ ΓN ∪ {0}, ξ1 6= 1 and ξd 6= 0, we have Z I[0,1] (ξ1 , . . . , ξd ) =

d ^

1>t1 >···>td >0 j=1

ωξj (tj ),

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where for all j = 1, . . . , d

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ωξj (tj ) =

   

dtj , if ξj = 6 0; − tj dtj , if ξj = 0. tj

ξj−1

  

Recall that an element x in any coalgebra (C, , ∆) is called group-like (see Definition A.6) if ∆(x) = x ⊗ x and (x) = 1. Proposition 13.3.13. For any path γ ∈ C∗ \ ΓN , we may define X Iγ := Iγ (ξ1 , . . . , ξd )xξ1 · · · xξd , d∈N0 ,ξ1 ,...,ξd ∈ΓN ∪{0}

where Iγ (∅) = 1. Then Iγ is a group-like element in (ChXi, •, ∆). Proof. Let X∗ be the set of words over X. For any word w = l1 . . . ln ∈ X∗ and any subset J = {i1 , . . . , ik } of {1, . . . , n} in increasing order, we define w|J = li1 . . . lin . Further set w|∅ = 1. Since ∆ is an algebra homomorphism we have n Y (lj ⊗ 1 + 1 ⊗ lj ) ∆(w) = ∆(l1 ) . . . ∆(ln ) = j=1

X

=

(w|J1 ) ⊗ (w|J2 ) =

J1 ∪J2 ={1,...,n},J1 ∩J2 =∅

by the definition of This implies that ∆(Iγ ) =

(w, u1


u )u 2

1

u1 ,u2 ∈X∗

⊗ u2


, where (w, u
u ) is the coefficient of w in u
u . 1

X n∈N0 ,ξ1 ,...,ξn ∈ΓN ∪{0} m∈N0 ,τ1 ,...,τm ∈ΓN ∪{0}

=

X

X n∈N0 ,ξ1 ,...,ξn ∈ΓN ∪{0} m∈N0 ,τ1 ,...,τm ∈ΓN ∪{0}

2

 Iγ ξ

1


τ



2

xξ1 · · · xξn ⊗ xτ1 · · · xτm

Iγ (ξ)Iγ (τ )xξ1 · · · xξn ⊗ xτ1 · · · xτm ,

by Lemma 2.1.2(iv), where ξ = (ξ1 , . . . , ξn ) and τ = (τ1 , . . . , τm ). Thus ∆(Iγ ) = Iγ ⊗ Iγ as desired. For any 0 < α < z < 1 and ξ1 , . . . , ξn ∈ ΓN ∪ {0}, we can also prove I[α,z] (ξ1 , . . . , ξn ) is group-like in the following way. Fixing α and regarding I[α,z] as a function of z, we see that it provides a solution of the differential equation X d I = Ω I where Ω = ωξ (z)xξ . ξ∈ΓN ∪{0}

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Note that Ω is primitive for the coproduct ∆ so both ∆ I[α,z] and I[α,z] ⊗ I[α,z] are the solutions to the differential equation dJ = ∆(Ω)J satisfying the initial condition J(α) = 1 ⊗ 1. Therefore ∆ I[α,z] = I[α,z] ⊗ I[α,z] . Proposition 13.3.14. For all z ∈ (0, 1), the series X I[0,z] (ξ1 , . . . , ξd )xξ1 · · · xξd (13.13) I[0,z] := lim πY (I[α,z] ) = α→0

ξ1 ,...,ξd ∈ΓN ∪{0} ξd 6=0,d∈N0

is a group-like element in (ChhYii, ∆). Proof. Regarding α as a variable in (0, 1), by Prop. 13.3.13 we see that the series πY (I[α,z] ) is a group-like element in (C(0,1) hhY ii, ∆) since πY is a coalgebra homomorphism. Taking α → 0 we see that the limit πY (I[0,z] ) exists by Prop. 2.3.8. This completes the proof of the proposition since taking limit is also a coalgebra homomorphism. For any word ys1 ,ξ1 ys2 ,ξ2 · · · ysd ,ξd over Y, we define the endomorphism q of ChYi as q(ys1 ,ξ1 ys2 ,ξ2 · · · ysd ,ξd ) = ys1 ,ξ1 ys2 ,ξ2 /ξ1 · · · ysd ,ξd /ξd−1 .

(13.14)

It has an inverse defined by p(ys1 ,ξ1 ys2 ,ξ2 · · · ysd ,ξd ) = ys1 ,ξ1 ys2 ,ξ1 ξ2 · · · ysd ,ξ1 ξ2 ···ξd .

(13.15)

Remark 13.3.15. The map q originates from the formula I[0,1] ({0}s1 −1 , µi1 , . . . , {0}sd −1 , µid ) = LN (s1 , . . . , sd |i1 , i2 −i1 , . . . , id −id−1 ) where (s1 , µi1 ) 6= (1, 1). Its inverse p gives rise to the operator tog of Eq. (8.24). Proposition 13.3.16. For any z ∈ (0, 1), we have the following identity in ChhYii: X Lz := Lis (zξ1 , ξ2 , . . . , ξd )ys1 ,ξ1 ys2 ,ξ2 · · · ysd ,ξd = q(I[0,z] ). d∈N0 ,s∈Nd ξ1 ,...,ξd ∈ΓN

Proof. This is just a restatement of Prop. 2.3.8. In the next result, we need to apply the projection πcv to remove the divergent terms from Eq. (13.13).

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Proposition 13.3.17. The series X Icv := lim− πcv I[0,z] = Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 06/12/16. For personal use only.

z→1

ξ1 ,...,ξd ∈ΓN ∪{0},ξ1 6=1,ξd 6=0

I[0,1] (ξ1 , . . . , ξd )xξ1 · · · xξd

is a group-like element in ChhXiicv . Proof. Prop. 13.3.16 shows that as z → 1− the asymptotic expansion of every coefficient of the series πcv (I[0,z] ) is given by the associated power
) series (see Eq. (2.9)) of Li(M (ξ) M ≥1 ∈ DivLogN for some s ∈ Nd and s ξ ∈ (ΓN )d ((s1 , ξ1 ) 6= (1, 1)), which converges by Cor. 2.3.10. The proposition now follows from Abel’s lemma on the convergence of a series at the boundary of its convergence interval (i.e., Lemma 2.3.7). By applying πcv we can remove the divergent terms as follows: X Lis (ξ1 , . . . , ξd )ys1 ,ξ1 · · · ysd ,ξd . lim πcv Lz = z→1−

d∈N0 ,s=(s1 ,...,sd )∈Nd ξ1 ,...,ξd ∈ΓN ,(s1 ,ξ1 )6=(1,1)

By straightforward computation we find the following crucial relation between the above two elements. Proposition 13.3.18. In ChhYiicv one has Lcv := lim πcv Lz = q(Icv ). z→1−

Proof. Noticing that q and πcv commute, by the definition of πcv we get Lcv = lim− πcv Lz = lim− πcv q(I[0,z] ) z→1

z→1

(by Prop. 13.3.16)

= q( lim− πcv I[0,z] ) = q(Icv ) z→1

by Prop. 13.3.17. Proposition 13.3.19. The series Lcv is a group-like element in ChhYiicv . Proof. First we observe that Y :=

X

yn,ξ

(13.16)

n∈N0 ,ξ∈ΓN

is a group-like element in (ChhYii, •, ∆∗ ) (see Exercise 13.5). Next it can be checked easily that for any λ ∈ C, both of the maps yn,ξ 7→ λn yn,ξ and yn,ξ 7→ ξyn,ξ are homomorphisms of the coalgebra (ChhYii, ∆∗ ), meaning they commute with ∆∗ . Therefore by applying the first map once and the

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second m times to the sum in Eq. (13.16) we get a group-like element for ∆∗ (see Exercise 13.7): X ξ m λn yn,ξ . Ym (λ) := n∈N0 ,ξ∈ΓN

Now for any M ∈ N, we may define the M th partial sum of CMZVs by (cf. Eq. (2.13)) X

) Li(M s1 ,...,sd (ξ1 , . . . , ξd ) :=

M ≥k1 >···>kd >0

ξ1k1 · · · ξdkd . k1s1 · · · kdsd

Then it is straightforward to check that its generating series X ) Li(M L(M ) := s1 ,...,sd (ξ1 , . . . , ξd )ys1 ,ξ1 · · · ysd ,ξd

(13.17)

d∈N0 ,ξ1 ,...,ξd ∈ΓN s1 ,...,sd ∈N

can be expressed as (see Exercise 13.8)  1   1  1 1 L(M ) = YM YM −1 · · · Y2 Y1 . M M −1 2 1

(13.18)

Since each factor on the right-hand side is group-like we see that L(M ) must be group-like, too. We now can check that y1,1 ChhYii is a co-ideal for ∆∗ since ∆∗ (y1,1 ) = y1,1 ⊗ 1 + 1 ⊗ y1,1 by the definition. Therefore, by Prop. A.10 in Appendix A, the coproduct ∆∗ descends onto the quotient ChhYii/y1,1 ChhYii which is identified with ChhYiicv . As the projection πcv is clearly a coalgebra homomorphism (i.e., commutes with ∆∗ ) we see immediately that Lcv = lim πcv (L(M ) ) M →∞

is a group-like element in (ChhYii, ∆∗ ). This completes the proof of the proposition. The following two lemmas are immediate consequences of Cor. B.11 in Appendix B (see Exercise 13.10). Lemma 13.3.20. Every group-like element Φcv in (QhXicv , ∆) is the image of some group-like elements in (QhXi, ∆) under πcv . Moreover, any two such elements Φ1 and Φ2 in (QhXicv , ∆) are related by Φ2 = exp((λ2 − λ1 )x1 )Φ1 exp((ρ2 − ρ1 )x0 ) where λj and ρj are the coefficients of x1 and x0 of Φj (j = 1, 2), respectively.

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Proof. Apply Cor. B.11 twice. Lemma 13.3.21. Any group-like series Φcv of (QhhYiicv , ∆∗ ) is the image of a group-like element of (QhhYii, ∆∗ ). Moreover, any two such elements Φ1 and Φ2 are related by Φ2 = exp((λ2 − λ1 )y1,1 )Φ1 , where λ1 and λ2 are the coefficients of y1,1 in Φ1 and Φ2 respectively. With the above lemma available, we can define Definition 13.3.22. Let L (resp. I) be the unique group-like element in (ChhYii, ∆∗ ) (resp. (ChhXii, ∆)) such that πcv (L) = Lcv (resp. πcv (I) = Icv ) and in which the coefficients of y1,1 is 0 (resp. in which the coefficients of x0 and x1 are 0). We also call L (resp. I) the regularization of Lcv (resp. Icv ) . Remark 13.3.23. The coefficients of L (resp. I) may be viewed as regularized values of the series Lis1 ,...,sn (ξ1 , . . . , ξn ) (resp. the iterated integral I[0,1] (ξ1 , . . . , ξn )). Note that Racinet’s choice is different from that given by Ihara, Kaneko and Zagier in which the coefficient of y1,1 is T instead of 0. In what follows we will derive a relation between the two series L and I. The main tools are contained in the three subsections of Sec. 2.3. Suppose f : k1 −→ k2 is a C-linear map between two rings over C. If Φ ∈ k1 hhXii then we denote by f¯(Φ) the element in k2 hhXii obtained by applying f on the coefficients of Φ. By C-linearity we see that f¯ is a homomorphism of ChhXii-bimodules. Further, if f is an algebra homomorphism and Φ is group-like for ∆ then so is f¯(Φ). The same is true for Y and ∆∗ . By Cor. 2.3.10 all the coefficients of the series L(M ) defined by Eq. (13.17) are in DivLogN . By Remark 2.3.6 we see that As is an algebra homomorphism, so Lt := As(L(M ) ) is group-like in (C[t]hhYii, ∆∗ ), where we have extended As from CN to CN hhYii. Clearly, πcv Lt = Lcv since all convergent coefficients result in constant functions. Hence we can regard the space ChhYii as a subspace of C[t]hhYii. Lemma 13.3.24. In C[t]hhYii, we have
Lt = exp (t + γ)y1,1 L .

(13.19)

Proof. Noticing that the coefficient of y1,1 in Lt is Ash = t + γ (see Example 2.3.3), we can readily prove the lemma by applying Lemma 13.3.21.

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Lemma 13.3.25. The group-like element exp T y1,1 L in C[T ]hhYii is the generating series of L∗ (s|i; T ). Namely, X
exp T y1,1 L = L∗ (s|i; T )ys1 ,µi1 · · · ysd ,µid . d∈N0 ,s=(s1 ,...,sd )∈Nd i=(i1 ,...,id )∈(Z/N Z)d

Proof. By the proof of Thm. 4.1.6 we know the generating series of L∗ (s|i; T ) is group-like for ∆∗ since L∗ (s|i; T ) satisfies the stuffle relations. Then we can apply Lemma 13.3.21 to deduce our result since L∗ (1|0; T ) = T by Prop. 13.3.8. Corollary 13.3.26. Let h(k) = k



(M )

Li{1}k {1}k


 M ≥1

. Then the associated

power series Fp(h(k)) = (Li1 ) /k!, where Li1 (z) = − log(1 − z). Moreover, Ash(k) (t) has degree k.
(M ) Proof. By the definition, the coefficient of yk1,1 in L(M ) is Li{1}k {1}k . So the last sentence of the corollary follows immediately from Lemma 13.3.24. By the definition of Eq. (2.9), the function Fp(h(k)) is given by the power series fk (z) :=

X n1 >n2 >···>nk

z n1 . n1 n2 · · · nk

A simple differentiation yields the recurrence relation fk0 (z) = fk−1 (z)/(1 − z) for all k ≥ 1 and f0 (z) = 1. Now by induction and the fact that fk (0) = 0 for all k > 0, we get immediately that f (z) = (Li1 (z))k /k!, as desired. Proposition 13.3.27. The coefficient of xk1 in the group-like element I[0,z] is (Li1 (z))k /k!. Proof. By the definition, the coefficient is k Z 1−z  k Z z dt dt = − 1−t t 0 1 by substitution t → 1 − t. This clearly evaluates to (Li1 (z))k /k!. Proposition 13.3.28. There exists S(y1,1 ) ∈ C[[y1,1 ]] such that L = S(y1,1 )qπY (I).

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Proof. Define It = As(I[0,z] ) which is a group-like element of (C[t]hhYii, ∆) so that πcv It = Icv . Since the coefficient of x1 in It is AsLi1 (t) = t we conclude from Lemma 13.3.21 that It = exp(x1 t)πY (I).

(13.20)

On the other hand, by Prop. 13.3.16 Lz = q(I[0,z] ) we find cp(Lt ) = q It by using the commutative diagram on page 19. Since cp is an endomorphism of ChhYii-modules

we get from Eqs. (13.19) and (13.20) that L = cp exp (2t + γ)y1,1 qπY (I). Now that

all the coefficients of L and I are in C we see that cp exp (2t + γ)y1,1 must be in C[[y1,1 ]]. This completes the proof of the proposition.
Lemma 13.3.29. The group-like element exp T x1 I in C[T ]hhXii is the generating series of L
(s|i; T ). Namely, X
exp T x1 I = L
(s|i; T )x0s1 −1 xµi1 · · · xs0d −1 xµid . d∈N0 ,s=(s1 ,...,sd )∈Nd i=(i1 ,...,id )∈(Z/N Z)d

Proof. By the proof of Prop. 13.3.13 we know the generating series of L
(s|i; T ) is group-like for ∆ since L
(s|i; T ) satisfies the shuffle relations. Then we can apply Lemma 13.3.20 to deduce our result since L
(1|0; T ) = T by Prop. 13.3.8. Next, we want to determine S explicitly. Theorem 13.3.30. The relation between L and I is given by L = S(y1,1 ) · qπY (I), ! ∞ X (−1)n−1 ζ(n) n where S(u) = exp u . n n=2 Proof. This follows easily from Thm. B.7 in Appendix B. Let’s recall the linear map defined in Eq. (3.3.17): ρ : R[T ] → R[T ],

ρ(eT u ) 7→ A(u)eT u ,

where A(u) = S(u)−1 . Let L0 := exp(T y1,1 ) · L and I0 := exp(T x1 ) · I. By Lemma 13.3.21, the coefficients of y1,1 and x1 in L0 and I0 are both equal to T . Note that y1,1 = x1 . For any s = (s1 , . . . , sn ) ∈ Nn and i = (i1 , . . . , in ) ∈ (Z/N Z)n , by Lemma 13.3.25 (resp. Lemma 13.3.29)

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L∗ (s|i; T ) (resp. L
(s|i; T )) is the coefficient of ys1 ,µi1 · · · ysn ,µin (resp. xs01 −1 xµi1 · · · x0sn −1 xµin , in L0 (resp. I0 ). Finally we prove the regularized DBSFs as given in Thm. 13.3.9 which can be reformulated as follows. Theorem 13.3.31. For any indices s1 , . . . , sn ∈ N and i1 , . . . , in ∈ Z/N Z, we have
ρ L∗ (s1 , . . . , sn |i1 , i2 − i1 , . . . , in − in−1 ; T ) = L
(s1 , . . . , sn |i1 , . . . , in ; T ). Proof. Dividing both sides of the equation in Thm. 13.3.30 by S(y1,1 ), we obtain A(y1,1 ) · L0 = exp(T y1,1 ) · A(y1,1 ) · L = exp(T y1,1 ) · qπY (I)

= qπY (exp(T x1 ) I) = qπY (I0 ).

Hence ρ(L0 ) = ρ(exp(T y1,1 ) · L) = A(y1,1 ) · exp(T y1,1 ) · L = qπY (I0 ). Finally, comparing the coefficient of ys1 ,µi1 ys1 ,µi2 −i1 . . . ysn ,µin −in−1 on both sides of the equation yields the theorem quickly. 13.3.3

Weight One Relations

If the level N > 3 then there are some non-trivial linear relations in CMZV(1, N ) of weight one whose structure is known to us, which we now review briefly. Fix the N th root of unity µ = exp(2πi/N ) as above. We know the cyclotomic units can be constructed using 1 − µj for 1 ≤ j < N and we have X µk − log(1 − µ) = = Li1 (µ). k k>0

Here we have chosen the branch cut of the logarithm by deleting the nonpositive real numbers (−∞, 0] from the complex plane. The next lemma is straightforward. Lemma 13.3.32. All of the following relations hold for Li1 . (i) (Symmetry relations). For all 1 < j < N/2, Li1 (µj ) − Li1 (µ−j ) =


(N − 2j)πi N − 2j = Li1 (µ) − Li1 (µ−1 ) . N N −2

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(ii) (Distribution relations). For any divisor d of N and 1 ≤ a < d, we have X Li1 (µa+dj ) = Li1 (µaN/d ). 0≤j