The Wilson function transform
Wolter Groenevelt
TL;DR
This work constructs two unitary Wilson-function transforms with kernels given by Wilson functions, extending Wilson polynomials to a non-polynomial spectral framework. By exploiting a duality involution on parameters, the authors establish complete spectral decompositions into continuous and discrete parts and provide explicit transform formulas, including the action on Wilson polynomials and Jacobi-function relations. The approach unifies a Dirichlet-type spectral analysis with reproducing-kernel methods and reveals deep connections to Jacobi function transforms and the q→1 limit of Askey–Wilson theory. The results offer explicit mapping properties, duality, and potential representation-theoretic interpretations (e.g., Racah coefficients) for Wilson functions and polynomials. Overall, the paper delivers a thorough spectral-theoretic construction of Wilson-function transforms and explicit transformation formulas for fundamental orthogonal systems therein.
Abstract
Two unitary integral transforms with a very-well poised $_7F_6$-function as a kernel are given. For both integral transforms the inverse is the same as the original transform after an involution on the parameters. The $_7F_6$-function involved can be considered as a non-polynomial extension of the Wilson polynomial, and is therefore called a Wilson function. The two integral transforms are called a Wilson function transform of type I and type II. Furthermore, a few explicit transformations of hypergeometric functions are calculated, and it is shown that the Wilson function transform of type I maps a basis of orthogonal polynomials onto a similar basis of polynomials.