Orthogonal polynomials and coherent pairs: the classical case

AbstractLet {Pn(x)}n ≥ 0 and {Rn(x)}n ≥ 0 be two sequences of simple monic polynomials such that (∗) Pn(x) = 1n + 1R′n + 1(x) − σnR′n(x), n = 0, 1, 2, … where {σn}n ≥ 0 is a sequence of complex numbers. Consider the two following problems: 1.(i) if {Rn}n ≥ 0 is a given system of orthogonal polynomials, to characterize all the sequences of orthogonal polynomials {Pn}n ≥ 0 and all the sequences of compatible parameters {σn}n ≥ 0 for which (∗) holds;2.(ii) the analogous problem, with the assumption that {Pn}n ≥ 0 is the given system of orthogonal polynomials. The first problem has been partially solved by Iserles et al. in [6], in the case in which {Rn}n ≥ 0 is a classical family. Here, we characterize the solution for both problems in the case in which the given system is some classical one