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# $q$-Integral and Moment Representations for $q$-Orthogonal Polynomials

Published:2002-08-01
Printed: Aug 2002
• Dennis Stanton
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## Abstract

We develop a method for deriving integral representations of certain orthogonal polynomials as moments. These moment representations are applied to find linear and multilinear generating functions for $q$-orthogonal polynomials. As a byproduct we establish new transformation formulas for combinations of basic hypergeometric functions, including a new representation of the $q$-exponential function $\mathcal{E}_q$.
 Keywords: $q$-integral, $q$-orthogonal polynomials, associated polynomials, $q$-difference equations, generating functions, Al-Salam-Chihara polynomials, continuous $q$-ultraspherical polynomials
 MSC Classifications: 33D45 - Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) 33D20 - unknown classification 33D2033C45 - Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions] 30E05 - Moment problems, interpolation problems