Hahn–Exton q-Bessel function

Koelink and Swarttouw proved that ${\displaystyle J_{\nu }^{(3)}(x;q)}$ has infinite number of real zeros. They also proved that for ${\displaystyle \nu >-1}$ all non-zero roots of ${\displaystyle J_{\nu }^{(3)}(x;q)}$ are real (Koelink and Swarttouw (1994)). For more details, see Abreu, Bustoz & Cardoso (2003). Zeros of the Hahn-Exton q-Bessel function appear in a discrete analog of Daniel Bernoulli's problem about free vibrations of a lump loaded chain (Hahn (1953), Exton (1983))

For the (usual) derivative and q-derivative of ${\displaystyle J_{\nu }^{(3)}(x;q)}$, see Koelink and Swarttouw (1994). The symmetric q-derivative of ${\displaystyle J_{\nu }^{(3)}(x;q)}$ is described on Cardoso (2016).

The Hahn–Exton q-Bessel function has the following recurrence relation (see Swarttouw (1992)):

${\displaystyle J_{\nu +1}^{(3)}(x;q)=\left({\frac {1-q^{\nu }}{x}}+x\right)J_{\nu }^{(3)}(x;q)-J_{\nu -1}^{(3)}(x;q).}$

The Hahn–Exton q-Bessel function has the following integral representation (see Ismail and Zhang (2018)):

${\displaystyle J_{\nu }^{(3)}(z;q)={\frac {z^{\nu }}{\sqrt {\pi \log q^{-2}}}}\int _{-\infty }^{\infty }{\frac {\exp \left({\frac {x^{2}}{\log q^{2}}}\right)}{(q,-q^{\nu +1/2}e^{ix},-q^{1/2}z^{2}e^{ix};q)_{\infty }}}\,dx.}$

${\displaystyle (a_{1},a_{2},\cdots ,a_{n};q)_{\infty }:=(a_{1};q)_{\infty }(a_{2};q)_{\infty }\cdots (a_{n};q)_{\infty }.}$

The Hahn–Exton q-Bessel function has the following hypergeometric representation (see Daalhuis (1994)):

${\displaystyle J_{\nu }^{(3)}(x;q)=x^{\nu }{\frac {(x^{2}q;q)_{\infty }}{(q;q)_{\infty }}}\ _{1}\phi _{1}(0;x^{2}q;q,q^{\nu +1}).}$

This converges fast at ${\displaystyle x\to \infty }$. It is also an asymptotic expansion for ${\displaystyle \nu \to \infty }$.