A q-Hankel transform associated to the quantum linking groupoid for the quantum SU(2) and E(2) groups

TL;DR

This paper develops a q-analogue of Erdelyi's Hankel transform formula using the quantum linking groupoid connecting quantum SU(2) and E(2) groups, involving q-Bessel functions and Wall polynomials.

Contribution

It introduces a new q-Hankel transform associated with the quantum linking groupoid for quantum SU(2) and E(2), extending classical integral transforms to the quantum group setting.

Findings

Derived a q-analogue of Erdelyi's formula for the Hankel transform.

Expressed the kernel of the q-Hankel transform via the 1 extphi1-q-Bessel function.

Calculated the transform of a product of Wall polynomials and q-exponentials as a product of Wall polynomials and q-exponentials.

Abstract

A q-analogue of Erdelyi's formula for the Hankel transform of the product of Laguerre polynomials is derived using the quantum linking groupoid between the quantum SU(2) and E(2) groups. The kernel of the q-Hankel transform is given by the 1\varphi1-q-Bessel function, and then the transform of a product of two Wall polynomials times a q-exponential is calculated as a product of two Wall polynomials times a q-exponential.