Jacobi operator, q-difference equation and orthogonal polynomials
Abstract
In this paper, a link between -difference equations, Jacobi operators and orthogonal polynomials is given. Replacing the variable by in a Sturm-Liouville -difference equation we discovered the Jacobi operator. With appropriate initial conditions, the eigenfunctions of such operators are either -orthogonal polynomials or the modified -Bessel function and a newborn the -Macdonald ones. The new Polynomial sequence we found is related to the -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 -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
Cite
@article{arxiv.1211.0359,
title = {Jacobi operator, q-difference equation and orthogonal polynomials},
author = {Lazhar Dhaouadi and Mohamed Jalel Atia},
journal= {arXiv preprint arXiv:1211.0359},
year = {2012}
}