Vol. 252, No. 2, 2011

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Connection relations and expansions

Mourad E. H. Ismail and Mizan Rahman

Vol. 252 (2011), No. 2, 427–446
Abstract

We give new proofs of the evaluation of the connection relation for the Askey–Wilson polynomials and for expressing the Askey–Wilson basis in those polynomials using q-Taylor series. This led to some inverse relations. We also evaluate the coefficients in the expansions of (x + b)n in various q-orthogonal polynomials, including the Askey–Wilson polynomials, which leads to explicit expressions for the moments of the Askey–Wilson weight function. We generalize the q-plane wave expansion by expanding q(x;α) in Askey–Wilson polynomials. Further, we prove a bibasic extension of the Nassrallah–Rahman integral and establish a recently conjectured identity of George Andrews.

Keywords
connection relations, bibasic integrals, moments of the Askey–Wilson and q-ultraspherical distributions, q-plane wave expansions, bibasic integrals, Andrews conjecture
Mathematical Subject Classification 2000
Primary: 05A19, 33D15
Secondary: 33D70
Milestones
Received: 29 June 2010
Revised: 28 March 2011
Accepted: 28 March 2011
Published: 29 October 2011
Authors
Mourad E. H. Ismail
Department of Mathematics
City University of Hong Kong
Tat Chee Avenue
Kowloon
Hong Kong
Department of Mathematics
King Saud University
Riyadh
Saudi Arabia
Mizan Rahman
Department of Mathematics
Carleton University
Ottawa, Ontario K1S5B6
Canada