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arXiv · 1208.2521

On q-analogues of the Fourier and Hankel transforms

Tom H. Koornwinder·René F. Swarttouw
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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.

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Tom H. Koornwinder, René F. Swarttouw. 2012-08-13. On q-analogues of the Fourier and Hankel transforms. https://arxiv.org/abs/1208.2521

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