On discrete coherent pairs of measures

In Castillo and Mbouna [On another extension of coherent pairs of measures, Indag. Math. 31 (2020), pp. 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 <= 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.

Automated summary

In this paper, a new concept of coherent pairs is introduced and known results are revisited using the discrete Hahn difference operator. Additionally, a detailed description of self-coherent pairs in specific situations is provided. The classical orthogonal polynomials are shown as special cases of a four-parametric family of q-polynomials.