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 (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 ): For a contour 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