On 𝑞-analogues of the Fourier and Hankel transforms

Koornwinder, Tom H.; Swarttouw, René F.

doi:10.1090/S0002-9947-1992-1069750-0

On $q$-analogues of the Fourier and Hankel transforms
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by Tom H. Koornwinder and René F. Swarttouw PDF

Trans. Amer. Math. Soc. 333 (1992), 445-461 Request permission

Abstract:

For H. Exton’s $q$-analogue of the Bessel function (going back to W. Hahn in a special case, but different from F. H. Jackson’s $q$-Bessel functions) we derive Hansen-Lommel type orthogonality relations, which, by a symmetry, turn out to be equivalent to orthogonality relations which are $q$-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 ($q$-cosines and $q$-sines which admit $q$-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.

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