Ismail M., Koelink E. (editors) Theory and Applications of Special Functions

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282

THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS

(iii) does not occur in Theorem 4.1, since it is assumed that kl

k2. If

we no longer assume this, the discrete terms in case (iii) correspond to

a finite number of positive discrete series representations.

The finite number of negative discrete series in Theorem 4.1 can be

obtained

described in Remark 5.28. In order to obtain the Clebsch-

Gordan coefficients in this case from the summation formula in Theorem

3.1, we need to consider different overlap coefficients for the continuous

series representations. Let vn(x;

u-,(x;

-E,

y), where

un is the Meixner-Pollaczek function as defined by (2.5). Then

is a generalized eigenvector of

~"9"

(X9),

see (Koelink,

54.4.11). From

Theorem 3.1, with (kl,k2,x1,x2,p)

I+

(k2,k17x2,x1, -p) we find the

Clebsch-Gordan coefficients for the eigenvector

(21

12)

these Clebsch-Gordan coefficients are multiples of

continuous

Hahn func-

tions

In this case the discrete mass points in the measure for the continuous

Hahn transform correspond to one complementary series representation,

or a finite number of negative discrete series representations.

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THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS

Vilenkin,

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NEW PROOFS OF

SOME Q-SERIES RESULTS*

Mourad

E.H.

Ismail

Department of Mathematics

University of Central Florida

Orlando,

3281

Ruiming Zhang

Department of Mathematics

University of

South

Florida

Tampa,

33620-5700

Abstract

We use a Sheffer classification technique to give very short proofs of the

addition theorem for the

function, the representation of

as a mul-

tiple of a

291,

and a relatively new representation of

E,.

A direct proof

of the evaluation of the connection coefficients of the Askey-Wilson

polynomials and the Nassrallah-Rahman integral are also given. A sim-

ple proof of a characterization theorem for the continuous q-Hermite

polynomials is also given.

Keywords:

Sheffer classification, delta operators, polynomial bases, Askey-Wilson

operators, the q-exponential function

Introduction

By a polynomial sequence

{

fn(x)} we mean a sequence of polynomials

such that fn(x) is a polynomial of exact degree

in x. Let

be a linear

operator defined on polynomials and reduces the degree of a polynomial

hence

is a nonzero constant. Given such operator one can

inductively define a polynomial sequence {fn(x)) such that

'Research partially supported

NSF

grant

DMS

99-70865.

This work was done at the

University of South Florida

O 2005

Springer Science+Business Media, Inc.

286

THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS

We say that a polynomial sequence {fn(x)) with fo(x)

belongs to

an operator T if (T

fn)

(x)

fn-l(~). It is known that given a sequence

of polynomials {f,(x)) with

of precise degree n then there exists an

operator

of the form

such that ak(x) has degree at most

and

f,(x)

fn-l

(x). For details,

see (Rainville, 1971, Chapter 13).

Theorem

.l.

Assume that two polynomial sequences

{r,

(x)) and

{s,

(x))

belong to the same operator T.

Then there is a sequence of constants

{a,) with a0

such that

Conversely if (1.2) holds with a0

and {r,(x)) belongs to

then

{s,(x)) also belongs to

Theorem 1.1 is implicit in the Sheffer classification (Rainville, 1971),

the umbra1 calculus (Rota et al., 1973) and the q-umbra1 calculus (Ismail,

2001) where additional assumptions are made but these assumptions do

not enter in the proof of (1.2). In 92 we apply Theorem

to give few

line proofs of three identities whose original proofs are lengthy.

Throughout this work we shall follow the notations and terminology

in (Andrews et al., 1999) and (Gasper and Rahman, 1990) for basic

hypergeometric series.

Theorem

1.1

will be applied where T is related to the the

Askey-

Wilson operator Dq which we now define.

Given a function

we set

(e")

(x), x

cos8, that is

In other words we think of

(cos 8) as a function of eie. The Askey-

Wilson divided difference operator Dq, (Ismail, 1995), (Askey and Wil-

son, 1985, p. 32), is defined by

It easy to see that the action of Dq on Chebyshev polynomials is given

Series and Integrals

hence Vq reduces the degree of a polynomial by one and

For applications of Vq to deriving summation theorems see (Ismail, 1995)

and (Cooper, 1996).

The polynomial bases we will be concerned with are

(cos 8; a)

(ae", ae

-ie;

pn (COS 8)

e2")

(-q

2-ne2i0;

q2)

,-in0

n-1

In the calculus of the Askey-Wilson operator the bases {cpn(x;a)),

{P~(x)), play the role played by ((1

2ax

u~)~), and

((1

x)~), re-

spectively, in the differential and integral calculus. On the other hand

{pn(x)), and {un(x, y)) play the role of {xn) and {(x

Y)~), respec-

tively, with y

cos

For expansions of entire functions in the first

three bases in (1.4) we refer the interested reader to (Ismail and Stan-

ton, 2003b).

The q-exponential function of Ismail and Zhang (Ismail and Zhang,

Ismail and Zhang (Ismail and Zhang, 1994) also defined

q-exponential function by

(1.5)

a two variable

It is evident from (1.5) that the function lq(x; t) is entire in x for all t,

It1

and is analytic in

for It1

and all x.

In $3 we give a lemma, Lemma 3.1, which is a special case of the

Nassrallah-Rahman integral (Nassrallah and Rahman, 1985) by using a

288

THEORY AND APPLICATIONS

SPECIAL FUNCTIONS

change of base formula from (Ismail, 1995). Lemma 3.1 and q-integration

by parts (Brown et al., 1996) are used to solve the connection coefficient

problem for Askey-Wilson polynomials. The original proof of the con-

nection coefficients is in (Askey and Wilson, 1985). In

we show that

Lemma 3.1 and the Watson transformation imply the full Nassrallah-

Rahman result.

In (Al-Salam, 1995) W. Al-Salam proved the following characteriza-

tion theorem:

Theorem

1.2.

Let

{pn(x))

be a polynomial sequence,

po(x)

{pn(x))

are orthogonal polynomials and

A(t)

is a formal power series

then

pn(x)

is a multiple of the

Hermite polynomials, (Askey and Ismail,

1983).

we use Theorem

1.1

and Corollary 2.1 together with properties

of continuous q-Hermite polynomials to give a simple proof of Theorem

1.2. We outline

derivation of every formula used, so the developments

presented here are self-contained.

We hope that the elementary nature of the proofs presented here will

lead to a better understanding of the subject matter.

Theorem

1.1

and

its

Applications

First note that given a sequence of polynomials {pn(x)) with pn(x)

of degree

and po(x)

we can define a linear operator T on the

vector space of polynomials by Tpo(x)

and Tpn(x)

pn-1 for

Hence

always exists. For representation of

as a differential

operator, possibly of infinite order, see Theorem 70 of $122 in Rainville

(Rainville, 1971).

Proof of Theorem

.I.

Let

Thus

Series and Integrals

289

Hence an,k+l

an-l,k,

01.

an,k+l

an-k-l,O and (1.2) follows. Con-

versely given (1.2) construct the operator

to which {rn(x)) belongs.

It follows from (1.2) that {sn(x)) belongs to

and the result follows.

The following corollary is a restatement of Theorem

1.1

in the lan-

guage of formal power series.

Corollary

2.1.

Two polynomial sequences {rn(x)), and {sn(x)), with

ro(x)

so(x)

belong to the same operator

if and only

where A(t)

antn with a0

and an is independent of x.

n=O

Ismail and Stanton (Ismail and Stanton, 2003a) proved

and

,2iB

-k

(eos

e2")

;

qb2/4tke-ibB

(2.3)

k=O

(4; q)k

e2ieq-k)

Suslov (Suslov, 2001) proved the following addition theorem for

290

THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS

belong to the operator Vq. Thus Corollary 2.1 and (1.5)) after replacing

t by tq1I4, shows that there is a power series B(t) such that

Both series defining the above

2p1

and Eq(x; t) are analytic functions of

t for It1

and are entire functions of x. Moreover Eq(x; t) and the

291

in (2.8) are equal to

at t

0. Hence B(t) is analytic in a neighborhood

of t

0. Substituting x

(q1/4

q-1/4) /2 we see that

where we used the q-binomial theorem in the last step.

simplification

completes the proof.

Note that (2.2) implies

Thus

Eq(O; t)

Proof of

(2.3)

and

(2.4). The relationships (2.6) and (2.7) imply that

belong to Vq for all

Thus there are formal power series A(t) and B(t)

such that

qn2/4

Ep (x;

A(t)

(4.; q)n

(XI tn

n=O

Series and Integrals 291

This establishes (2.3) and (2.4) since pn(0)

6n,0, Eq(O,y; t)

&(y;t)

imply A(t)

and

(t)

Eq (y

;

Connection Coefficients

Ismail (Ismail, 1995) proved a q-Taylor series expansion for polyno-

mials in cpn(x; a) and applied it to derive the q-analogue of the Pfaff-

Saalschiitz theorem in the form

~n(x;

(9; qln (b/a)k

(abqk,

bla;

n-k

c~k(x; a).

(3.1)

]C=O (Q; q)k (4; q)n-k

Let

(,2ie,

.-2ie;

(cos 8;

where

stands for the vector (al, a2, as, a4). The Askey-Wilson integral

is (Askey and Wilson, 1985)

(e2ie, e-2ie; q), dB

1(al,m,as,a4)

(ajeie, aje-ie; q),

j=1

have the evaluation

Lemma

(e2ie, e-2ie; q),(aeie, ae-ie; q), d8

(ajeie,

q),

j=1