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On \(q\)-analogues of the Fourier and Hankel transforms. (English) Zbl 0759.33007

The \(q\)-Bessel function treated here is a \({_ 1\Phi_ 1}\) with numerator parameter equal to zero. This is a limit of the \(q\)-Jacobi polynomials introduced by Hahn. The authors derive a \(q\)-version of the Hankel transform from Hahn’s orthogonality of his \(q\)-Jacobi polynomials, obtain a \(q\)-extension of Craf’s addition formula, point out that it is also an analogue of the Weber-Schafheitlin integral, and point out that attractive anlogues of Fourier cosine and sine transformations are included in the \(q\)-Hankel transforms. In addition, a wise comment on notation is included at the end of the first section.
Reviewer: R.Askey (Madison)

MSC:

33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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