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.
Koelink and Swarttouw proved that has infinite number of real zeros.
They also proved that for all non-zero roots of 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))
Derivatives
For the (usual) derivative and q-derivative of , see Koelink and Swarttouw (
1994). The symmetric q-derivative of is described on Cardoso (
2016).
Recurrence Relation
The Hahn–Exton q-Bessel function has the following recurrence relation (see Swarttouw (
1992)):
Alternative Representations
Integral Representation
The Hahn–Exton q-Bessel function has the following integral representation (see
Ismail and Zhang (
2018)):
Hypergeometric Representation
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 .
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.
Koelink and Swarttouw proved that has infinite number of real zeros.
They also proved that for all non-zero roots of 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))
Derivatives
For the (usual) derivative and q-derivative of , see Koelink and Swarttouw (
1994). The symmetric q-derivative of is described on Cardoso (
2016).
Recurrence Relation
The Hahn–Exton q-Bessel function has the following recurrence relation (see Swarttouw (
1992)):
Alternative Representations
Integral Representation
The Hahn–Exton q-Bessel function has the following integral representation (see
Ismail and Zhang (
2018)):
Hypergeometric Representation
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 .