Connection relations and expansions

Mourad E. H. Ismail and Mizan Rahman

Vol. 252 (2011), No. 2, 427–446

DOI: 10.2140/pjm.2011.252.427

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 ℰ_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