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.
is the basic hypergeometric function.
has infinite number of real zeros.
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
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)): This converges fast at