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 . 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
 
 
 
 
 
 
 
 
 

Special functions
Q-analogs