On discrete coherent pairs of measures

TL;DR

This paper extends the concept of coherent pairs of measures to discrete orthogonal polynomials using the Hahn difference operator, providing a comprehensive classification of self-coherent pairs and addressing a conjecture by Ismail.

Contribution

It introduces a discrete framework for coherent pairs using the Hahn difference operator and classifies self-coherent pairs, linking classical and q-orthogonal polynomials.

Findings

Describes discrete self-coherent pairs for specific parameters.

Rewrites known results within the Hahn difference operator framework.

Provides a partial answer to Ismail's conjecture on orthogonal polynomials.

Abstract

In [Castillo \& Mbouna, Indag. Math. {\bf 31} (2020) 223-234], the concept of $\pi_N$-coherent pairs of order $(m,k)$ with index $M$ is introduced. This definition, implicitly related with the standard derivative operator, automatically leaves out the so-called discrete orthogonal polynomials. The purpose of this note is twofold: first we use the (discrete) Hahn difference operator and rewrite the known results in this framework; second, as an application, we describe exhaustively the (discrete) self-coherent pairs in the situation whether $M=0$, $N\leq2$, and $(m,k)=(1,0)$. This is proved by describing in a unified way the classical orthogonal polynomials with respect to Jackson's operator as special or limiting cases of a four parametric family of $q$-polynomials. This gives a partial answer to a conjecture posed by M. E. H Ismail in his monograph [Classical and quantum orthogonal polynomials in one variable, Cambridge University Press, 2005].