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Transactions of the American Mathematical Society

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On $ q$-analogues of the Fourier and Hankel transforms

Authors: Tom H. Koornwinder and René F. Swarttouw
Journal: Trans. Amer. Math. Soc. 333 (1992), 445-461
MSC: Primary 33D45; Secondary 33D80, 39A10, 44A15
MathSciNet review: 1069750
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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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