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 (Pn)n and (Qn)n whose derivatives of higher orders m and k (resp.) are connected by a linear algebraic structure relation such as N i=0 ri,n P(m) n−i+m(x) = M i=0 si,n Q (k) n−i+k(x) for all n = 0, 1, 2, . . ., where M and N are fixed nonnegative integer numbers, and ri,n and si,n are given complex parameters satisfying some natural conditions. Let u and v be the moment regular functionals associated with (Pn)n and (Qn)n (resp.). Assuming 0 m k, we prove the existence of four polynomials ΦM+m+i and ΨN+k+i , of degrees M +m+i and N +k + i (resp.), such that Dk−m(ΦM+m+iu) = ΨN+k+iv (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.