On discrete coherent pairs of measures. Issue 7 (3rd July 2022) - Álvarez-Nodarse, R.; Castillo, K.; Mbouna, D.; Petronilho, J.
Abstract : In Castillo and Mbouna [On another extension of coherent pairs of measures, Indag. Math. 31 (2020), pp. 223–234], 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.