Hahn–Exton q-Bessel function

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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

is the basic hypergeometric function.

Properties[edit]

Zeros[edit]

Koelink and Swarttouw proved that has infinite number of real zeros (Koelink and Swarttouw (1994)).

Derivative[edit]

For the (usual) derivative of , see Koelink and Swarttouw (1994).

Recurrence Relation[edit]

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

Alternative Representations[edit]

Integral Representation[edit]

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

Hypergeometric Representation[edit]

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

This converges fast at . It is also an asymptotic expansion for .

References[edit]