On 𝑞-analogues of the Fourier and Hankel transforms

For H. Exton’s q q -analogue of the Bessel function (going back to W. Hahn in a special case, but different from F. H. Jackson’s q q -Bessel functions) we derive Hansen-Lommel type orthogonality relations, which, by a symmetry, turn out to be equivalent to orthogonality relations which are q q -analogues of the Hankel integral transform pair. These results are implicit, in the context of quantum groups, in a paper by Vaksman and Korogodskiĭ. As a specialization we get ( q q -cosines and q q -sines which admit q q -analogues of the Fourier-cosine and Fourier-sine transforms. We also get a formula which is both an analogue of Graf’s addition formula and of the Weber-Schafheitlin discontinuous integral.