The Askey-Wilson function transform

Abstract

In this paper we present an explicit (rank one) function transform which contains several Jacobi-type function transforms and Hankel-type transforms as degenerate cases. The kernel of the transform, which is given explicitly in terms of basic hypergeometric series, thus generalizes the Jacobi function as well as the Bessel function. The kernel is named the Askey-Wilson function, since it provides an analytical continuation of the Askey-Wilson polynomial in its degree. In this paper we establish the $L^2$-theory of the Askey-Wilson function transform, and we explicitely determine its inversion formula.

LaTeX2e file. 19 pages

Cited by in corpus (6)

• Properties of generalized univariate hypergeometric functions
• Koornwinder polynomials and the XXZ spin chain
• Macdonald difference operators and Harish-Chandra series
• Bispectral extensions of the Askey-Wilson polynomials
• Connection problems for quantum affine KZ equations and integrable lattice models
• An expansion formula for the Askey-Wilson function