On the zeros of the Hahn-Exton q-Bessel function and associated q-Lommel polynomials

Abstract

For the Bessel function \begin{equation} \label{bessel} J_ν(z) = \sum\limits_{k=0}^{\infty} \frac{(-1)^k \left( \frac{z}{2} \right)^{ν+2k}}{k! Γ(ν+1+k)} \end{equation} there exist several $q$-analogues. The oldest $q$-analogues of the Bessel function were introduced by F. H. Jackson at the beginning of this century, see M. E. H. Ismail \cite{Is1} for the appropriate references. Another $q$-analogue of the Bessel function has been introduced by W. Hahn in a special case and by H. Exton in full generality, see R. F. Swarttouw \cite{Sw1} for a historic overview. Here we concentrate on properties of the Hahn-Exton $q$-Bessel function and in particular on its zeros and the associated $q$-Lommel polynomials.