An integral representation and some transformation properties of q-Bessel functions

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Abstract

A Poisson-type integral representation for Jackson's q-Bessel function is obtained by using Askey and Wilson's q-beta integral and Nassrallah and Rahman's integral formula for an 8ϑ7 series. This representation along with some transformtion formulas for basic hypergeometric series help express the q-Bessel functions as a 3ϑ2 series in base q, a 2ϑ2 series in base q2 and a 2ϑ1 series in base qq12, where 0 < q < 1. As an application, q-analogues are found for Gegenbauer's degenerate addition formulas for Bessel functions.

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This work was supported by the NSERC under Grant A6197.