Hahn–Exton q-Bessel function

In mathematics, the Hahn–Exton q-Bessel function or the third Jackson q-Bessel function is a q-analog of the Bessel function, introduced by Hahn (1953) in a special case and by Exton (1983) in general.

The Hahn–Exton q-Bessel function is given by

$J_\nu^{(3)}(x;q) = \frac{x^\nu(q^{\nu+1};q)_\infty}{(q;q)_\infty} \sum_{k\ge 0}\frac{(-1)^kq^{k(k+1)/2}x^{2k}}{(q^{\nu+1};q)_k(q;q)_k}$