Jacobi operator, q-difference equation and orthogonal polynomials

2012/11/02 by Dhaouadi, Lazhar, Atia, Mohamed Jalel
#FOS: Mathematics #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.1211.0359

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

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