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

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second addition formula for continuous q-ultraspherical polynomials

353

Rational biort hogonal functions for the

Askey-Roy q-beta measure

The

Askey-Roy q-beta integral

(see (Askey and Roy, 1986, (3.4)),

(Gasper, 1987), (Askey, 1988, pp. 92, 93) (Gasper and Rahman, 1990,

Exercise 6.17(ii))) is as follows:

Here the

q-gamma function

is defined by

The special case

a+l

of (3.1) goes back (without proof) to Ramanu-

jan in Chapter 16 of his second notebook (see (Berndt, 1991, p. 29, Entry

14)), and later in his paper (Ramanujan, 1915, (19)), with subsequent

proof by Hardy (Hardy, 1915).

When we let

in (3.1) then we formally obtain the

beta integral

on (0,oo):

When we move the orthogonality relations

(3.3)

PP@)

(x)

Pk'P)

(x) (1

x)"

x)P dx

of the

Jacobi polynomials

354

THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS

(see (Erdelyi et al., 1953, $10.8)) to [O,oo) by the substitution

t)/(l

t), then we obtain orthogonality relations

Thus the rational functions t

I-+

P?")

((1

t)/(l

t)),

Z>o, are

orthogonal with respect to the beta measure on

[0,

oo) of which the

total mass is given in (3.3). We would like to find q-analogues of these

orthogonal rational functions such that the orthogonality measure is the

q-beta measure in (3.1).

From (1.15) we have

Observe from (3.1) that, for k,

Z20,

Thus

second addition formula for continuous q-ultraspherical polynomials

355

Proof of Theorem 1.2. Multiply both sides of (3.8) with the factor

(q-n,

q"+a+P+l;

and sum from

0 to

Then the right-hand

(4; 4)k

side becomes

by (Gasper and Rahman, 1990, (1.5.2)). Thus

where (3.10) is obtained by a similar argument as (3.9). Then (3.9) and

(3.10) together with (1.14) imply the biorthogonality relations (1.16).

The q-integral version of the Askey-Roy integral (3.1) is

Here

Rea, Rep

-1,

and the q-integral is defined by

(1.17). The case

qC of (3.11) (which is no real restriction) was

given in (Gasper, 1987) (see also (Askey, 1988, (2.27))) and in (Gasper

and Rahman, 1990, Exercise 6.17(i))). Another approach to (3.11) is

presented in (De Sole and Kac, 2003).

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THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS

For

any function on

(0,

oo)

for which the sum below converges ab-

solutely, we have:

3.00

(-sq-C, -s-lql+c; q) (-tqP+l, -t-lqff+2.

(t) tc-'

(-s, -s-1q. 74,

)

(-tq-c, -t-lql+y q), dqt

(3.12)

The right-hand side, and thus the left-hand side of (3.12) is independent

Henceforth we will take

0 without loss of information.

Then

(3.11) together with (3.12) takes the form

The second equality in (3.13) is Ramanujan's sum (Gasper and

Rahman, 1990, (5.2.1)). This observation is the usual way to prove

(3.11). With a completely analogous argument as used for the proof of

Theorem

1.2,

we can next prove Theorem 1.3. Details are omitted.

In completion of this section, observe the following symmetry of the

functions p?@)(t; q, rat):

q-n,

qn+a+P+l

p?J) (t; q, rat)

292

q"+l, -tqP+l

- -

(-1)"

(qP+l; q), tn (-t-lqa+l

;

pi~7a)(t-1; q, rat).

(qa+'

;

q)n (-tqP+l;

(3.14)

Here the first and last equality are just (1.14). We use (Gaper and

Rahman, 1990, (1.5.4)) for the second and fourth equality, while the

second addition formula for continuous q-ultraspherical polynomials

357

third equality is obtained by reversion of summation order in a termi-

nating q-hypergeometric series (see (Gasper and Rahman, 1990, Exercise

lA.(ii))).

Concluding

remarks

The results of this paper lead to several interesting questions. I formu-

late two of these questions here. I also discuss specializations of Theorem

1.2.

As I already mentioned in

$1,

the new addition formula in the case of

the continuous q-Legendre polynomials (formula (1.9)) was first obtained

in a quantum group context, where a two-parameter family of Askey-

Wilson polynomials, including the continuous q-Legendre polynomials,

was interpreted as spherical functions on the SUq(2) quantum group.

Here the left and right invariance of the spherical functions was no longer

with respect to the diagonal quantum subgroup, but infinitesimally with

respect to twisted primitive elements in the corresponding quantized

universal enveloping algebra. The

and

variables in the addition

formula (1.9) are parameters for the twisted primitive elements occurring

in the left respectively right invariance property. On the other hand, the

3(P2

factors on the right-hand side of (1.9), involving

resp.

can be

rewritten as the functions p$;)(.

;

q, rat) of argument -qFU resp. -9-*.

So I wonder whether an interpretation of these last functions and of

their biorthogonality (discussed in $14.3) can be given in the context of

su,

(2).

A second question is whether the biorthogonality relations for the

functions peyB)(t; q, rat) (Theorems 1.2 and 1.3) fit into a more general

class of biorthogonal rational functions. In fact, several papers have

appeared during the last 10 or 20 years which discuss explicit systems

of biorthogonal rational functions depending on many parameters and

expressed as q-hypergeometric functions, see Rahman (Rahman, 1986),

Wilson (Wilson, 1991), Ismail and Masson (Ismail and Masson, 1995)

and Spiridonov and Zhedanov (Spiridonov and Zhedanov, 2000). How-

ever, I did not see how the functions pp'P)(t; q, rat) and their biorthog-

onality relations can be obtained as special or limit case of families dis-

cussed in these references. If the functions pP@)(t; q, rat) are indeed

unrelated to the functions discussed in these references, then it is a nat-

ural question how to generalize the system of biorthogonal functions

pp." (t; q, rat).

On the other hand, our functions ppB)(t; q, rat) have some interesting

limit cases, one of which has occurred earlier in literature. When we take

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THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS

limits for

and/or

,L3

in (1.14), then we obtain:

p?@)(t;

rat)

191

(q-n; qa+l;

-qt)

(4.1)

phm.P)(t; q, rat)

191

(q-"; -tqP+l; q, -qt)

(4.2)

p~m~m)(t; q, rat)

(qMn;

q, -qt)

(4.3)

In (4.1) and (4.3) we have polynomials of degree

rather than

rational functions in t. The limit case

,L3

of the biorthogonality

relations (1.16) then becomes:

where

The further limit case

of the biorthogonality relations (4.4) then

becomes

q-c2

rq(c)

rq(i

(t; q, rat)

qc)

r(1

q) (q; q)m

tc-l dt (4.6)

(m+')(t-lq; q, rat)

Pm (-tq-c -t-1 '+c.

,dm

Similar limit cases can be considered for the biorthogonality relations

(1.3). The biorthogonality relations (4.6) are essentially the ones ob-

served by Pastro (Pastro, 1985, pp. 532, 533). He also points out that

biorthogonality relations of the form

Pn(t)Qm(t-l) dp(t)

6,,,

with

and Qn polynomials of degree

can be rewritten as orthog-

onality relations on the real line for Laurent polynomials. This is in-

deed the case in (4.6). Pastro also observes that the biorthogonality

measure occurring in (4.6) is a (non-unique) orthogonality measure for

the Stieltjes- Wigert polynomials (see (Koekoek and Swarttouw, 1998,

53.27)).

second addition formula for continuous q-ultraspherical polynomials

359

Note

For further discussion and references concerning the Ismail-Zhang

ex-

pansion of a generalized q-exponential function, mentioned in Remark

2.4, see $4.5 in

S. K. Suslov, An Introduction to Basic Fourier Series,

Kluwer, 2003.

References

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Padmanab-

han, K. S., and Thangaraj,

V.,

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and Wilson,

(1985). Some basic hypergeometric orthogonal polynomials

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York

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and Suslov, S. K. (1998). Basic analog of Fourier series on a q-quadratic

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W.,

Oberhettinger, F., and Tricomi, F. G. (1953). Higher tran-

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Gasper, G. (1984). Letter to

Askey dated July 23, 1984. Personal communication.

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bridge.

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(1915). Proof of a formula of Mr. Ramanujan. Messenger Math., 44:18-

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and Masson, D. R. (1995). Generalized orthogonality and continued

fractions.

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R.,

and Suslov, S. K. (1999). The q-Bessel function on

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and q-special functions (Montre'al,

QC,

1996), volume

of CRM Proc. Lecture

Notes, pages 183-200. Amer. Math. Soc., Providence, RI.

Ismail, M. E. H. and Zhang,

(1994). Diagonalization of certain integral operators.

Adv. Math., 109:l-33.

Koekoek, R. and Swarttouw,

F. (1998). The Askey-scheme of hypergeometric

orthogonal polynomials and its q-analogue. Report 98-17, Faculty of Technical

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http://aw.twi.tudelft.nl/"koekoek/askey/.

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Koelink, E.

(1997).

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son, D. R., and Rahman, M., editors, Special functions, q-series and related topics

(Toronto, ON, 1995), volume

of Fields Institute Communications, pages

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Amer. Math. Soc., Providence, RI. Electronic version:

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V. (2001).

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(1993).

Askey-Wilson polynomials as zonal spherical functions

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(1985).

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A BILATERAL SERIES INVOLVING

BASIC HYPERGEOMETRIC FUNCTIONS

almar Rosengren

Department of Mathematics

Chalmers University of Technology and Goteborg University

SE-412

Goteborg

SWEDEN

hjalrnar@rnath.chalrners.se

Abstract

We prove a summation formula for a bilateral series whose terms are

products of two basic hypergeometric functions.

special cases, series

of this type arise

matrix elements of quantum group representations.

Introduction

The object of the present paper is to study bilateral series of a type

that first appeared in the work of Koelink and Stokman (Koelink and

Stokman, 2001), in the context of harmonic analysis on the SU(1,l)

quantum group. They needed to compute sums of the form

which appeared as matrix elements of quantum SU(1,l) representations.

Several special cases were considered, with different, very technical,

proofs (some of these proofs were found by Mizan Rahman, see the

appendix to (Koelink and Stokman, 2001)).

In joint work with Koelink (Koelink and Rosengren, 2002), we gave

an extension of the summation formulas from (Koelink and Stokman,

2001), with a unified proof. Namely, we showed that under natural

conditions of convergence and the assumption

(and thus also

abx), the sum in (1.1) can be expressed as the sum

of two

sW7

series, or, alternatively, as the sum of three balanced

493

2005

Springer Science+Business Media, Inc.

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THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS

series. From the quantum algebraic viewpoint, which was only briefly

mentioned in (Koelink and Rosengren, 2002), this computes a general

class of matrix elements for the strange, the complementary and the

principal unitary series of quantum SU(1,l). The proof in (Koelink

and Rosengren, 2002) is similar to, though not an extension of, one of

Rahman's proofs in (Koelink and Stokman, 2001). In particular, it is

quite technical and involves rather non-obvious applications of q-series

transformations.

In a related paper (Stokman, 2003), Stokman considered matrix ele-

ments for the principal unitary series and gave a very simple proof for

their expression as

8W7

series.

The idea is simple but powerful: the

representations are realized on

L2(T),

where one wants to compute the

scalar product of two particular functions. Expanding both functions as

Fourier series gives a special case of the sum

(1.1).

Stokman's observa-

tion, which is our starting point, is that the integral defining the scalar

product can be computed very easily using residue calculus.

Our aim is to use Stokman's method not only to give a simple proof

of the summation formula in (Koelink and Rosengren, 2002), but also

to give a far-reaching extension of this identity.

Dedication

This paper would not have been written without the pioneering con-

tributions of Mizan Rahman. I'm happy to know him, not only through

his work as a master of identities, but also as a genuinely kind and help-

ful person. It is a pleasure to dedicate this paper to him as a small token

of my appreciation.

bilateral summation

We will now state the main result of the paper. Throughout, we

use the standard notation of (Gasper and Rahman, 1990). The base

is assumed to satisfy 0

1q1

We denote by

.+lcp,

not only the

convergent basic hypergeometric series, but also its analytic continuation

It>1;

cf. (Gasper and Rahman, 1990,

94.5).

Theorem

2.1.

Let

It>o,

and let the other parameters below

satisfy