# 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, and satisfies the Hahn-Exton q-difference equation (Swarttouw (1992)). This function was introduced by Hahn (1953) in a special case and by Exton (1983) in general.

The Hahn–Exton q-Bessel function is given by

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

${\displaystyle \phi }$ is the basic hypergeometric function.

## Properties

### Zeros

Koelink and Swarttouw proved that ${\displaystyle J_{\nu }^{(3)}(x;q)}$ has infinite number of real zeros (Koelink and Swarttouw (1994)).

### Derivative

For the (usual) derivative of ${\displaystyle J_{\nu }^{(3)}(x;q)}$, see Koelink and Swarttouw (1994).

### Recurrence Relation

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

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

## Alternative Representations

### Integral Representation

The Hahn–Exton q-Bessel function has the following integral representation (see Ismail and Zhang (2016)):

${\displaystyle J_{\nu }^{(3)}(z;q)={\frac {z^{\nu }}{\sqrt {\pi \log q^{-2}}}}\int _{-\infty }^{\infty }{\frac {\exp \left({\frac {x^{2}}{\log q^{2}}}\right)}{(q,-q^{\nu +1/2}e^{\mathrm {i} x},-q^{1/2}z^{2}e^{\mathrm {i} x};q)_{\infty }}}\mathrm {d} x.}$
${\displaystyle (a_{1},a_{2},\cdots ,a_{n};q)_{\infty }:=(a_{1};q)_{\infty }(a_{2};q)_{\infty }\cdots (a_{n};q)_{\infty }.}$

### Hypergeometric Representation

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 ${\displaystyle \nu \to \infty }$.