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. This function was introduced by in a special case and by in general.
The Hahn–Exton q-Bessel function is given by
is the basic hypergeometric function.

Properties

Zeros

Koelink and Swarttouw proved that has infinite number of real zeros.
They also proved that for all non-zero roots of are real. For more details, see and. 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

Derivatives

For the derivative and q-derivative of, see. The symmetric q-derivative of is described on.

Recurrence Relation

The Hahn–Exton q-Bessel function has the following recurrence relation :

Alternative Representations

Integral Representation

The Hahn–Exton q-Bessel function has the following integral representation :
For a contour integral representation, see.

Hypergeometric Representation

The Hahn–Exton q-Bessel function has the following hypergeometric representation :
This converges fast at. It is also an asymptotic expansion for.