A counterexample to a conjecture of M. Ismail

TL;DR

This paper provides a counterexample that disproves the second part of a conjecture by M. Ismail regarding the characterization of certain orthogonal polynomials, clarifying the limits of the conjecture's applicability.

Contribution

It presents a specific counterexample that conclusively disproves the second part of Ismail's conjecture on polynomial characterization.

Findings

Counterexample disproves the second part of Ismail's conjecture

The first part of the conjecture remains valid

The issue of the conjecture is now definitively closed

Abstract

In an earlier work [K. Castillo et al., J. Math. Anal. Appl., 514 (2022) 126358], we give positive answer to the first, and apparently more easy, part of a conjecture of M. Ismail concerning the characterization of the continuous $q$-Jacobi polynomials, Al-Salam-Chihara polynomials or special or limiting cases of them. In this note we present an example that disproves the second part of such a conjecture, and so this issue is definitively closed.