On linearly related sequences of derivatives of orthogonal polynomials

Abstract

We discuss an inverse problem in the theory of (standard) orthogonal polynomials involving two orthogonal polynomial families ( P n ) n and ( Q n ) n whose derivatives of higher orders m and k (resp.) are connected by a linear algebraic structure relation such as ∑ i = 0 N r i , n P n − i + m ( m ) ( x ) = ∑ i = 0 M s i , n Q n − i + k ( k ) ( x ) for all n = 0 , 1 , 2 , … , where M and N are fixed nonnegative integer numbers, and r i , n and s i , n are given complex parameters satisfying some natural conditions. Let u and v be the moment regular functionals associated with ( P n ) n and ( Q n ) n (resp.). Assuming 0 ⩽ m ⩽ k , we prove the existence of four polynomials Φ M + m + i and Ψ N + k + i , of degrees M + m + i and N + k + i (resp.), such that D k − m ( Φ M + m + i u ) = Ψ N + k + i v ( i = 0 , 1 ) , the ( k − m ) th-derivative, as well as the left-product of a functional by a polynomial, being defined in the usual sense of the theory of distributions. If k = m , then u and v are connected by a rational modification. If k = m + 1 , then both u and v are semiclassical linear functionals, which are also connected by a rational modification. When k > m , the Stieltjes transform associated with u satisfies a non-homogeneous linear ordinary differential equation of order k − m with polynomial coefficients.