On discrete coherent pairs of measures

Abstract

In [Castillo & Mbouna, Indag. Math. 31 (2020) [223][224][225][226][227][228][229][230][231][232][233][234], the concept of π 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 ≤ 2, 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.