On q-analogues of the Fourier and Hankel transforms

TL;DR

This paper develops q-analogues of Fourier and Hankel transforms using third q-Bessel functions, establishing orthogonality relations and integral transform pairs that extend classical harmonic analysis into the quantum group setting.

Contribution

It introduces new q-analogues of Fourier and Hankel transforms based on third q-Bessel functions, including orthogonality relations and addition formulas.

Findings

Derived Hansen-Lommel type orthogonality relations for third q-Bessel functions

Established q-analogues of Fourier-cosine and Fourier-sine transforms

Presented a formula combining Graf's addition and Weber-Schafheitlin integrals

Abstract

For the third q-Bessel function (first introduced by F.H. Jackson, later rediscovered by W.Hahn in a special case and by H. Exton) we derive Hansen-Lommel type orthogonality relations, which, by a symmetry, turn out to be equivalent to orthogonality relations which are q-analogues of the Hankel integral transform pair. These results are implicit, in the context of quantum groups, in a paper by Vaksman and Korogodskii. As a specialization we get q-cosines and q-sines which admit q-analogues of the Fourier-cosine and Fourier-sine transforms. We get also a formula which is both an analogue of Graf's addition formula and of the Weber-Schafheitlin discontinuous integral. This is a corrected version of a paper which appeared in Trans. Amer. Math. Soc. 333 (1992), 445-461.