Some Basic Lommel Polynomials

Abstract

The basic Lommel polynomials associated to the₁ϕ₁q-Bessel function and the Jacksonq-Bessel functions are considered as orthogonal polynomials inq^ν, whereνis the order of the corresponding basic Bessel functions. The corresponding moment problems are both indeterminate and determinate depending on a parameter. Using techniques of Chihara and Maki we derive an explicit orthogonality measure, which is discrete and unbounded. For the indeterminate moment problem this measure is N-extremal. Some results on the zeros of the basic Bessel functions, both as functions of the order and of the argument are obtained. Precise asymptotic behaviour of the zeros of the₁ϕ₁q-Bessel function is obtained.