Jacobi operator, q-difference equation and orthogonal polynomials

Abstract

In this paper, a link between $q$-difference equations, Jacobi operators and orthogonal polynomials is given. Replacing the variable $x$ by $ q^{-n}$ in a Sturm-Liouville $q$-difference equation we discovered the Jacobi operator. With appropriate initial conditions, the eigenfunctions of such operators are either $q$-orthogonal polynomials or the modified $q$-Bessel function and a newborn the $q$-Macdonald ones. The new Polynomial sequence we found is related to the $q$-Lommel polynomials introduced by Koelink and other. Adapting E. C. Titchmarsh's theory, we showed the existence of a solution square-integrable only in the complex case. As application in the real case we gave the behavior at infinity for $q$-Macdonald's function. Finally, we pointed out that the method described in our paper can be generalized to study the orthogonal polynomial sequence introduced by Al-Salam and Ismail