An inverse problem in the theory of (standard) orthogonal polynomials involving two orthogonal polynomial families and whose derivatives of higher orders and (resp.) are connected by a linear algebraic structure relation such as
for all , where and are fixed nonnegative integer numbers, and and are given complex parameters satisfying some natural conditions. Let and be the moment regular functionals associated with and (resp.). Assuming , we prove the existence of four polynomials and , of degrees and (resp.), such that
the 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 , then and are connected by a rational modification. If , then both and are semiclassical linear functionals, which are also connected by a rational modification. When , the Stieltjes transform associated with satisfies a non-homogeneous linear ordinary differential equation of order with polynomial coefficients.