Koornwinder, Tom H.; Swarttouw, René F. On \(q\)-analogues of the Fourier and Hankel transforms. (English) Zbl 0759.33007 Trans. Am. Math. Soc. 333, No. 1, 445-461 (1992). 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) Cited in 7 ReviewsCited in 91 Documents 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 Keywords:\(q\)-Bessel functions; \(q\)-Fourier transform; \(q\)-Hankel transform PDFBibTeX XMLCite \textit{T. H. Koornwinder} and \textit{R. F. Swarttouw}, Trans. Am. Math. Soc. 333, No. 1, 445--461 (1992; Zbl 0759.33007) Full Text: DOI arXiv