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

Подождите немного. Документ загружается.

q-Analogues of Some Multivariable Biorthogonal Polynomials

201

However, the above

291

equals

(4.13)

which vanishes unless

M. This completes the proof of (1.22).

Also, with

(3.6))

(4.6) and (4.10) give

(4.14)

which is, of course, the same as (1.23). By taking

2, e.g., in which

case the series on the right hand side of (4.14) becomes a terminating

balanced

4p3

series, it is easily seen that in general the above inner

product do& not vanish when

M and

#m.

In closing this section we would like to point out that unlike the q

case that corresponds to the Tratnik biorthogonalities, the q-analogues

Q,,

do not seem to work out the same way

Pm.

Transformations of

F,,,(x,

and

G,,,(x,

We shall now address the problem of proving the biorthogonality re-

lation (1.24). First of all, it is very simple to use (Gasper and Rahman,

1990a, (11.20)) to prove that

202

THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS

The forms of

Fm,,(x,

and

Gm,,(x,

that turn out to be most conve-

nient for the summations in

(1.24)

are

follows:

Since

q-Analogues of Some Multivariable Biorthogonal Polynomials

203

and

by (Gasper and Rahman, 1990a) (111.15))) (5.3) follows from (1.26) with

a bit of simplification.

To derive (5.2) from (1.25) we need two applications of (Gasper and

Rahman, 1990a, (111.15)) on each of the two

4.3

series involved in (1.25).

First

j-m-n

q-ml

gqx-N-1

4.3

[

74")

x+y-N-m-n+j

;

474

q-~,

dqx-N-1

j-m-n

4.3

[

ffc

wm,

;

q'q

Substituted into (1.25) this leads to another balanced

4.3

series:

204

THEORY AND APPLICATIONS

SPECIAL FUNCTIONS

which, when transformed twice in the same manner as in

(5.4),

leads to

After some simplifications

(5.4)

and

(5.5)

give

(5.2).

Denoting the left

hand side of

(1.24)

F,,,

G,I,,I,

it follows that

q-Analogues of Some Multivariable Biorthogonal Polynomials

where

Proof of (1.24)

Since each term in the weight function can be glued on nicely with the

x and

dependent terms of the two double series in (5.6), the x, y-sum

can be isolated as

by (Gasper and Rahman, 1990a, (11.21)). The sum over

(5.6) now reduces to

206

THEORY AND APPLICATIONS

SPECIAL FUNCTIONS

The sum over

is a multiple of

(Gasper and Rahman, 1990a, (1.9.10)). Together with a similar

ex-

pression for the sum over

we now have

q-Analogues of Some Multivariable Biorthogonal Polynomials

The

sum is

which vanishes unless m'

m and n'

The sum in (6.4), via (Gasper and Rahman, 1990a, (11.12) and (II.6)),

equals

which vanishes unless

m and n'

n. Thus we must have

Fm,n

Grnl,,1

unless (m, n)

(m', n')

and then

(6.5)

1-a

m+n

n-m

aq2m+2n

(~.l/c; q)m

(T',YE;~;

(0,

q-N; q)rn+n

which completes the proof of (1.24) and (1.28).

It may be mentioned that there are other double series representa-

tions for Fm,,(x,

that one could use instead of (5.2) in the derivation

of the biorthogonality relation (1.24) which do not contain the factor

(rN; q)z+y that cancels out the (q-N; q)%+. factor in the weight

function, but the subsequent computations turn out to be quite tedious,

while the final result is, of course, the same.

208

THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS

References

Gasper,

and Rahman,

(1990). Basic Hypergeometric Series. Cambridge Uni-

versity Press, Cambridge.

Gasper, G. and Rahman,

(2003). Some systems of multivariable orthogonal Askey-

Wilson polynomials. This Proceedings.

GranovskiY

and Zhedanov,

S. (1992). 'Twisted' Clebsch-Gordan coefficients

for su,(2).

Phys. A, 25:L1029-L1032.

Gustafson, R. A. (1987). A Whipple's transformation for hypergeometric series in

u(n)

and multivariable hypergeometric orthogonal polynomials. SIAM

Math.

Anal., 18:495-530.

Koelink,

and Van der Jeugt,

(1998). Convolutions for orthogonal polynomials

from Lie and quantum algebra representations. SIAM

Math. Anal., 29:794-822.

Rahman,

(1981). Discrete orthogonal systems corresponding to Dirichlet distribu-

tion. Utilitas Mathematzca, 20:261-272.

Rosengren,

(2001). Multivariable q-Hahn polynomials as coupling coefficients for

quantum algebra representations. Int.

Math. Sci., 28:331-358.

Tratnik,

V. (1991a). Some multivariable orthogonal polynomials of the Askey

tableau-continuous families.

Math. Phys., 32:2065-2073.

Tratnik,

V. (1991b). Some multivariable orthogonal polynomials of the Askey

tableau-discrete families.

Math. Phys., 32:2337-2342.

van Diejen, J.

and Stokman, J. V. (1998). Multivariable q-Racah polynomials.

Duke Math.

J.,

91:89-136.

Wilson, J. A. (1980). Some hypergeometric orthogonal polynomials. SIAM

Math.

Anal., 11:690-701.

SOME SYSTEMS OF MULTIVARIABLE

ORTHOGONAL ASKEY-WILSON

POLYNOMIALS

George Gasper

Department of Mathematics

Northwestern University

Evanston,

60208

Mizan Rahman*

School

Mathematics and Statistics

Carleton University

Ottawa, Ontario

CANADA

K1S

5B6

Abstract

In 1991 Tratnik derived two systems of multivariable orthogonal Wil-

son polynomials and considered their limit cases. q-Analogues of these

systems are derived, yielding systems of multivariable orthogonal Askey-

Wilson polynomials and their special and limit cases.

Introduction

In (Tratnik, 1991a) the Wilson (Wilson, 1980) polynomials

*Supported, in part,

NSERC grant

#A6197.

O 2005

Springer Science+Business Media, Inc.

210

THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS

were extended to the multivariable Wilson polynomials (in a different

notation)

where,

elsewhere,

These polynomials are of total degree Ns in the variables yl,

y, with

xi,

1,2,

. .

s, and they form a complete set for polynomials

in these variables.

In (Askey and Wilson, 1979) and (Askey and Wilson, 1985) the nota-

tions

(x2; a,

c, d) and

(-x2) are used for the polynomials in (1.1)

and their orthogonality relation is given. Tratnik (Tratnik, 1991a, (2.5))

proved that the W,(x) polynomials satisfy the orthogonality relation

for Re (a,

c, d, a2,.

,a,)

with

ix,)

(2ixs)

(-2ixS)