ORTHOGONAL POLYNOMIALS AND COHERENT PAIRS - THE CLASSICAL CASE

Let {P-n(x)}(n greater than or equal to 0) and {R(n)(x)}(n greater than or equal to 0) be two sequences of simple monic polynomials such that (*) P-n(x) = 1/n+1 R'(n+1)(x) - sigma(n)R'(n)(x), n=0,1,2,... where {sigma(n)}(n greater than or equal to 0) is a sequence of complex numbers. Consider the two following problems: (i) if {R(n)}(n greater than or equal to 0) is a given system of orthogonal polynomials, to characterize all the sequences of orthogonal polynomials {P-n}(n greater than or equal to 0) and all the sequences of compatible parameters {sigma(n)}(n greater than or equal to 0) for which (*) holds; (ii) the analogous problem, with the assumption that {P-n}(n greater than or equal to 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 greater than or equal to 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.