Epilegomena to the study of semiclassical orthogonal polynomials

TL;DR

This paper characterizes semiclassical orthogonal polynomials related to the Askey-Wilson operator, disproves a conjecture by Ismail, and explores the symmetry between different operators in the context of orthogonal polynomials.

Contribution

It provides new characterization theorems for $ ext{D}_q$-semiclassical orthogonal polynomials and presents counterexamples to Ismail's conjecture, enhancing understanding of their structure.

Findings

Disproved Ismail's conjecture with explicit counterexamples.

Established symmetry between standard derivative and Askey-Wilson operator.

Presented new characterization theorems for $ ext{D}_q$-semiclassical polynomials.

Abstract

In his monograph [Classical and quantum orthogonal polynomials in one variable, Cambridge University Press, 2005 (paperback edition 2009)], Ismail conjectured that certain structure relations involving the Askey-Wilson operator characterize proper subsets of the set of all $\mathcal{D}_q$-classical orthogonal polynomials, here to be understood as the Askey-Wilson polynomials and their limit cases. In this paper we give two characterization theorems for $\mathcal{D}_q$-semiclassical (and classical) orthogonal polynomials in consonance with the pioneering works by Maroni [Ann. Mat. Pura. Appl. (1987)] and Bonan, Lubinsky, and Nevai [SIAM J. Math. Anal. 18 (1987)] for the standard derivative, re-establishing in this context the perfect "symmetry" between the standard derivative and the Askey-Wilson operator. As an application, we present a sequence of $\mathcal{D}_q$-semiclassical orthogonal polynomials of class two that disproves Ismail's conjectures. Further results are presented for Hahn's operator.