Hypergeometric series recurrence relations and some new orthogonal functions

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ORTHOGONA HYPERGEOMETRIC SERIES RECURRENCE RELATIONS AND SOME NEW ORTHOGONA WILSON, JAMES JANIES ARTHUR ProQuest Dissertations and Theses; 1978; 1978; ProQuest

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' WILSON, James Arthur, 1951HYPERGEOMETRIC SERIES RECURRENCE RELATIONS AND SOME NEW ORTHOGONAL FUNCTIONS

The University of Wisconsin—Madison, Ph.D., 1978 Mathematics

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HYPERGEOME‘I‘RIC

ENCE RELATIONS SERIES RECURR

CTIONS ORTHOGONAL FUN AND SOME NEW te School of the ed to the Gradua A thesis submitt illment of in partial fulf consin—Madison University of Wis losophy of Doctor of Phi ts for the degree the requiremen 3!

WILSON JAMES ARTHUR

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S HYPEMBOMETRIC SERIES RECURRENCE RELATION AND SOME NEW ORTHOGONAL FUNCTIONS

A thesis submitted to the Graduate School of the

University of Wisconsin—Madison in partial fulfillment of

the requirements for the degree of Doctor of Philosophy

3!

ARTHUR WILSON——-—-— JAMES ____——— ______.__

Degree to be awarded:

Decanber 19

May 19

August 19.73..

Approved by Thesis Reading Committee:

(Mm Clo/55%

Major Professor

mm Is, Im

Date df Examination

Dean, Jeraduate School /

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HYPERGEOMETRIC SERIES RECURRENCE RELATIONS AND SOME NEW ORTHDGONAL FUNCTIONS

by JAMES ARTHUR WILSON

A thesis submitted in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY (Mathematics)

at the

UNIVERSITY OF WISCONSIN-MADISON

1978

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TABLE OF CONTENTS

. . . . . . ii

. . . .

.

. . . .

I.

INTRODUCTION. . . . . . . . . . . . .

.

. . . . . . .

II.

4173 POLYNOMIAL ORTHOGONALITIES. . . .

. . . .

. . . . . . . 14

. . . .

. . . .

. . . . . . . 31

THREE—TERM CONTIGUOUS RELATIONS . . . . . . .

. . . . . . . 43

ACKNOWLEDGMENTS. .

. . . . . . . .

III. GRAM DETERMINANTS . . . . . .

IV.

. . .

1

BIBLIOGRAPHY..........................62

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ACKNOWLEDGMENTS

For creating the atmosphere in which all the ideas in this thesis

sprang up inevitably, my gratitude goes to Professor Richard Askey. I also wish to thank Professors Dennis Stanton and George Andrews for

their valuable contributions to this environment, my parents and my wife Rosemary for their understanding and encouragement, and the fol-

lowing people for their inspiring examples:

Burl Cannon and Professors

Raymond Redheffet, Kirby Baker, Basil Gordon, Richard Arena, Alfred Hales, Theodore Motzkin, and Carl deBoor.

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I.

INTRODUCTION

This thesis is largely the result of efforts to better understand

the connection between hypergeometric series and orthogonal polynomials.

A hypergeomerric series is a series

(a1)k...(a )k

W

al,...,a ;

P

F p q(b1,...,bq

E (b) qk (1) k 1k ...(b) k—O

-

2)

2

k

with (a)k = a(a+1)...(a+k—1) if k 1 1, and (a)o = 1.

shifted factorial, since (1)k = M. and for ‘2] < 1 if q = p—l.

,

We call (a)k a

It converges for all z if q 1 p;

Orthogonal polynomials for which explicit

symbolic Calculations can be carried out due to the availability of

explicit formulas such as orthogonality relations, recurrence rela— tions, and differential or difference equations, seem invariably to involve hypergeometric series or q-series, generalizations of hyper—

geometric series.

The polynomials can be expressed as hypergeometric

series or q—series, and their properties are consequences of hypergeometric series theorems or their q—extensions.

This situation is

perhaps not so surprising, since a hypergeometric series is simply an

s

infinite series

2

t

k

with

t k+1 Itk a rational function of k, and

series more complicated than this are difficult to work with. (Excep—

tions are the q—series, which have tk+litk a rational function of qk. Many theorems for hypergeometric series generalize to q—series.)

However, what is more striking is that nearly all the special types of hypergeometric series for which summation or transformation

formulas exist are involved in an essential way with some orthogonal

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polynomials .

In the first chapter, we elaborate on these ideas while giving necessary background information on hypergeometric series and ortho-

General references for most of the orthogonal

gonal polynomials.

polynomial information are Erdélyi [5], volume 2, and Szegfi [11].

Hypergeometric series references are Bailey [1] and Erdélyi [5], volume 1.

In succeeding chapters, we introduce some new families

of orthogonal polynomials and biorthogonal rational functions whose

basic properties involve the deeper series identities, including some new three-series relations.

series.

We will not be concerned here with q-

Extensions to q-series are being worked out for all the new

results in this thesis. The orthogonal polynomials known most widely are the classical

polynomials named for Jacobi, Laguerre and Hermite, with the following

explicit representations and orthogonality relations: Jacobi Eolmmials:

((2,8) Pn (x)

_ _

(n+1) n!

n

-n,n+o+B+1; l—_x 2F1(o+l 2) ’

(1,3 > —1, n 1 0;

(1.1)

( e) (1:) dx - o, m 54 11 ; ( ) (x) rm“ f1 (l—x)a(1+x)BPnu’B -1

Laguerre nolflomials:

Lga>(n+s) , (2n+a+6)2

’

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(,5) d (x) 5911“

(1.3)

a

. n+u+8+1 (a+1,B+l) (x) ’ Pn—l 2

(1-x2)Cld—z' 2‘“ ”(2‘) + [s-a-(oc+8+2)x] E v§“'5)(x) + d2

(1.4)

+n(n+oc+5+1) 9(Q’B)(x) - 0; and

(3—1)“ —{(1-x°‘)*“(1+x)“*“}. (1—x)“(1+x)BP§“’B’(x) = 2n11! flat“

(1.5)

There are other orthogonal polynomials which make very nice dis—

crete analogs of the classical polynomials.

They satisfy similar

explicit formulas in which the derivative operator i: replaced by

the ordinary difference operator Af(x) = f(x+1) - f(x).

These poly-

nomials and their orthogonality relations are:

Hahn polynomials :

_

_

-n ,n+o,+8+1,-x-

Qn(x’°’s’m ' 3F2(u+1,-N

' 1) ’

a,S>-loro.,B0,0\—ix) .

We introduce in chapter two some new orthogonal polynomials, represented by AF3'S’ which include as limiting cases all the polynomials discussed so far and others as well.

Among the others are the

dual Hahn polynomials, obtained by interchanging n and x in Qu(x;o,B,N):

Rn(A(X);u.B.N)

=

(-n , -x , x+a+B+l ;

31’ 2 o.+1,—N

01,3 > --1 or 0.,5 < —N, 0 i n i N.

1) ’

These are polynomials of degree n in

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B the variable Mx)

-

x(x+a+8+l).

Ketlin and McGregor's orthogonality

relation [7] is

(1. 13)

N (mu)x (“g“)x(e.+1)x(-mx(—1)x 2: x=0 (1)x\(x);a,B,N)Rm(A(x);u.S,N)

0,

ma‘n.

The recurrence relation and difference equation for the Hahn polynomials are, respectively, the difference equation and recurrence

relation for their duals.

As consequences of properties of the 51:3

polynomials, the dual Hahn's satisfy some previously unnoticed ortho-

gonality relations (with conditions on the parameters different from

those above) and Rodrigues—type formulas. Special cases of the 4F3 polynomial orthogonalities have appeared

in the quantum mechanical theory of angular momentum as Racah's

orthogonality for 6-j symbols (see [9]).

However, the orthogonality

was recognized as a polynomial orthogonality only in very special

cases (Biedenharn et a1 [4], p. 253).

This is discussed further in

chapter two. We now consider the types of hypergeometric series which satisfy useful explicit formulas and mention some of the ways they tie in

with known orthogonal polynomials.

We are concerned mainly with 17+l

formulas, since the important formulas for other PF '5 are limiting

cases of these.

Examples:

ting a -> w gives (1.1h).

(1) Replacing x in (1.15) by x/a and let— (2) The Jacobi polynomial orthogonality, with

a change of variable, becomes an orthogonality for the polynomials

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9

2F1(;:in+a+5+1; x/B). orthogonality.

Letting B 9 +0 gives the Laguerre polynomial

(3) Szego ([11], section 8.1) shows how properties of

Bessel functions, 0171's, are consequences of properties of Jacobi polynomials . OFO I s and 1F0 u s are trivial..

(1.14)

03°C; x) - ex,

and

(1.15)

r (a; x) - (1-1:)‘3 .

l 0 -

These formulas give the total masses of the weight functions for the

Charlier, Meixner and Krawtchouk polynomials. 2171's, which are well represented among the polynomials listed,

satisfy quite a few important identities.

There are transformation

formulas called linear transformations, which include the symmetries

p“1“(1—p)“xn(N-x;1-p,m and

93"” (x) = (-1)“rt(15’°‘)(—x) There are quadratic transformations for 21’1'5 with various conditions

on the parameters.

One of these gives

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10

(Zn)!

(a,u)

(n+1)2n 1,2n

_ (x)

n!

_

(m,-2)

(u+l)n Pn

(2x2-l) '

sh; z), the hypergeometrit equation, The differential equation for 2F1(c’ is just (1.4) with a change of variable.

. a.b; 2) give

2Fl(c

(1.16)

Differentiation formulas for

(1.3) and

(1—x)“(1+x)31=§°"8)(x) =

-i i {(140 (1+1 (1+x) “11,211, 3+1) (1‘)} 211 dx

'

Iterating (1.16) n times yields the Rodrigues formula (1.5).

But the

formulas occurring most frequently among the polynomial properties are

Gauss's contiguous relations (see [6] or [5], volume 1) and simple con— sequences of them.

These are three-term identities which include

orthogonal polynomial recurrence relations and difference analogs of

(1.3), (1.4), and (1.16).

When 2 = 1, the ZFl can be evaluated by

Gauss ' s theorem:

(1'1”

.

Ewe—vb) 2F1(c3:1“; 1) _‘ i‘(c-a)I‘(c—-b)

(provided Re(c~a-b) > 0, so the series converges).

The terminating

series version,

-mb;

2F1(c

_

(c-b)

1) ’ 735‘“ 11

gives the total mass of the Hahn polynomial weight function. Most interesting 3F2 formulas are for series with z = 1, and

p+1FP formulas heyong this level require restrictions on the parameters.

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11 ao,al, . . . ,a ; P l) is called a k—balanced series if (1) The series 'p+1F1)b1,...,bp

b1+...+b —-a°a1-...-aP = k and one of ao,...,ap is a nonposnzive integer.

(Similar series without the terminating condition satisfy

more complicated identities which so far have not been useful.)

k = l, the series is called simply a balanced series.

called well—poised if a1 + bl =

= ap + bl, = a0 + 1.

If

The series is

It 15 very

well poised if, in addition, b1 - 190.

An extension of (1.17) needed in chapter two is the PEaff-Saal— schutz theorem:

(1.18)

—n,eh ,c; 1) P'(r1 3 2d

a

(d-b)n(d-C)n ——(d)n(d—b-C)n

,

provided the series is balanced.

The linear transformations and contiguous relations for ZFl'S generalize successively to 3 F 2 's with z = l, to balanced [aF 3 's, and to 2-balanced, very well poised 9FB' s.

(Note that the number of

free parameters increases by one at each stage.)

The 31:2 transfor—

mations contain the Hahn polynomial symmetry

(u+l)n Qn(x;u,B,N) = (-1)“ 1‘ (—N+s)

(a—(z‘i—1)N(a-cl+1)N

Then divide

As 5 —> 0, the integral term

0, and the result may be written

N

(2a) k(a+1) k (a+b)k(a+c) k(a+d)k

(23+1)N(l-c—d)N RED (1)k(s)k(a-b+l)k(a-c+l)k(a—d+1)k

(2‘11)

- pn((a+k)2)pm((a+k)2) n! (n+a+b+c+d—l)n (a+b)n (a+c)n(a+d) n (i:v+€:)n (134%)“ (c+d)n m,n

(a+b+c+d) 2n

Interchanging a and b here is equivalent to summing in the reverse

order.

(2.11) can also be proven directly from the 5174 formula (1.24)

just as (2.6) is proven from (2.3). Necessary and sufficient conditions on a,b,c,d for the positivity

of the weights in (2.11) are quite messy, but some sufficient conditions are

(2.12)

a+b = —N, b < )5 < a, -a < c < a+1, and either d > —b or d < b+1

Of course, interchanging a and b in (2.12) also gives sufficient

conditions for positivity. We now describe how, as claimed in chapter one, many orthogonality relations for previously known polynomials are included in the AFB or—

thogonalities as limiting cases.

The appropriate limit processes can

usually be determined by comparing the hypergeometrie series represent-

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24 It sometimes helps to write the AF3 poly—

ations of the polynomials.

nomials, with a change of variable and parameters, as

(2.13) with A(x)

_ rn(x(x),u.s.v.6)

=

-n n-i-u+B+1 -x x+v+6+l' ’ 1) 4F3(a+i,e+s+1.§+1’

x(x+'y+6+l).

=

Then (2 .11) becomes

(2. 14)

(TI-5+1) k ( y"3"3)k