An integral representation and some transformation properties of q-Bessel functions
As long ago as 1903, \textit{F. H. Jackson} [Trans. Roy. Soc. Edin. 41, 1-28 (1903)] introduced two q-analogues of the Bessel function of the first kind. The main object of this paper is to deduce a q-integral of Poisson type for the basic Bessel functions. In the analysis, the main tool used is a q-analogue of the Euler integral representation of the function \({}_ 2F_ 1\). (see \textit{B. Nassrallah} and \textit{M. Rahman}, SIAM J. Math. Anal. 16, 186-197 (1985; Zbl 0564.33009)]. In the sequel, quadratic transformations and q-analogues of certain results by Gegenbauer are obtained.
scientific article; zbMATH DE number 4032100
scientific article