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∑i=0Nri,nPn−i+m(m)(x)=∑i=0Msi,nQn−i+k(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 thatDk−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.