Expansion of analytic functions in q-orthogonal polynomials

Abstract

A classical result on the expansion of an analytic function in a series of Jacobi polynomials is extended to a class of q-orthogonal polynomials containing the fundamental Askey-Wilson polynomials and their special cases. The function to be expanded has to be analytic inside an ellipse in the complex plane with foci at ±1. Some examples of explicit expansions are discussed.Keywords Basic hypergeometric functions · q-orthogonal polynomials · Askey-Wilson polynomials · Continuous q-Jacobi polynomials · Continuous dual q-Hahn polynomials · Continuous q-ultraspherical polynomials · Al-Salam and Chihara polynomials · Continuous big q-Hermite polynomials · Continuous q-Hermite polynomials · Chebyshev polynomials · Jacobi polynomials Mathematics Subject Classification (2000) 33D45 · 42C10 · 33D15 Main resultsThe representation of an analytic function as a series involving polynomials is a fundamental problem in classical analysis and approximation theory; see for instance [9,13] and references therein. In the previous note [26] we have discussed an analog of the Cauchy-Hadamard formula, well-known for expansions in power series

On polynomial series expansions of Cliffordian functions

“…The representation of an analytic function as a series involving polynomials is a fundamental problem in classical analysis and approximation theory; see for instance [1] and [2] and references therein. In a series of papers, see for example [3][4][5], within the framework of Clifford analysis, this problem has been partially considered and studied.…”