Indeterminate moment problem associated with continuous dual q-Hahn polynomials

We study a limiting case of the Askey-Wilson polynomials when one of the parameters goes to infinity, namely continuous dual q-Hahn polynomials when q > 1. Solutions to the associated indeterminate moment problem are found and an orthogonality relation is established

A characterization of Askey-Wilson polynomials

Please read abstract in the article.The research of the first author was supported by a Vice-Chancellor’s Postdoctoral Fellowship from the University of Pretoria. The research by the second author was partially supported by the National Research Foundation of South Africa under grant number 108763.http://www.ams.org/publications/journals/journalsframework/prochj2019Mathematics and Applied Mathematic

Structure relations of classical orthogonal polynomials in the quadratic and q-quadratic variable

We prove an equivalence between the existence of the rst structure relation satis ed by a sequence of monic orthogonal polynomials fPng1n =0, the orthogonality of the second derivatives fD2 xPng1n =2 and a generalized Sturm{Liouville type equation. Our treat- ment of the generalized Bochner theorem leads to explicit solutions of the di erence equation [Vinet L., Zhedanov A., J. Comput. Appl. Math. 211 (2008), 45{56], which proves that the only monic orthogonal polynomials that satisfy the rst structure relation are Wilson poly- nomials, continuous dual Hahn polynomials, Askey{Wilson polynomials and their special or limiting cases as one or more parameters tend to 1. This work extends our previous result [arXiv:1711.03349] concerning a conjecture due to Ismail. We also derive a second structure relation for polynomials satisfying the rst structure relation.This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14).The research of MKN was supported by a Vice-Chancellor's Postdoctoral Fellowship from the University of Pretoria. The research by KJ was partially supported by the National Research Foundation of South Africa under grant number 108763.http://www.emis.de/journals/SIGMAam2019Mathematics and Applied Mathematic

On exponential and trigonometric functions on nonuniform lattices

We develop analogs of exponential and trigonometric functions (including the basic exponential function) and derive their fundamental properties: addition formula, positivity, reciprocal and fundamental relations of trigonometry. We also establish a binomial theorem, characterize symmetric orthogonal polynomials and provide a formula for computing the nth-derivatives for analytic functions on nonuniform lattices (q-quadratic and quadratic variables).This work was partially supported by the Alexander von Humboldt Foundation.https://link.springer.com/journal/111392020-05-01hj2019Mathematics and Applied Mathematic

On Solutions of Holonomic Divided-Difference Equations on Nonuniform Lattices

The main aim of this paper is the development of suitable bases that enable the direct series representation of orthogonal polynomial systems on nonuniform lattices (quadratic lattices of a discrete or a q-discrete variable). We present two bases of this type, the first of which allows one to write solutions of arbitrary divided-difference equations in terms of series representations, extending results given by Sprenger for the q-case. Furthermore, it enables the representation of the Stieltjes function, which has already been used to prove the equivalence between the Pearson equation for a given linear functional and the Riccati equation for the formal Stieltjes function. If the Askey-Wilson polynomials are written in terms of this basis, however, the coefficients turn out to be not q-hypergeometric. Therefore, we present a second basis, which shares several relevant properties with the first one. This basis enables one to generate the defining representation of the Askey-Wilson polynomials directly from their divided-difference equation. For this purpose, the divided-difference equation must be rewritten in terms of suitable divided-difference operators developed in previous work by the first author