Orthogonal polynomials and coherent pairs: the classical case

Abstract

Let { P n ( x)} n ≥ 0 and { R n ( x)} n ≥ 0 be two sequences of simple monic polynomials such that (∗) P n(x) = 1 n + 1 R′ n + 1(x) − σ nR′ n(x) , n = 0, 1, 2, … where { σ n } n ≥ 0 is a sequence of complex numbers. Consider the two following problems: 1. (i) if { R n } n ≥ 0 is a given system of orthogonal polynomials, to characterize all the sequences of orthogonal polynomials { P n } n ≥ 0 and all the sequences of compatible parameters { σ n } n ≥ 0 for which (∗) holds; 2. (ii) the analogous problem, with the assumption that { P n } 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 { R n } 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.