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 . 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 (usual) 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 (see ):
Alternative Representations Integral Representation The Hahn–Exton q-Bessel function has the following integral representation (see ):
Hypergeometric Representation The Hahn–Exton q-Bessel function has the following hypergeometric representation (see ):
This converges fast at . It is also an asymptotic expansion for .
References