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

${\displaystyle \phi }$ is the basic hypergeometric 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)):

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

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