On linearly related sequences of derivatives of orthogonal polynomials

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 [Graphics] 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 Phi(M+m+i) and Psi(N+k+i), of degrees M + m + i and N + k + i (resp.), such that Dk-m(Phi(M+m+i)u) = Psi(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. (C) 2008 Elsevier Inc. All rights reserved.