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On linearly related sequences of derivatives of orthogonal polynomials. (English) Zbl 1160.42011

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 1,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 0mk, we prove the existence of four polynomials Φ M+m-i and Ψ N+k+1 , of degrees M+m+1 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.

42C05General theory of orthogonal functions and polynomials
33C45Orthogonal polynomials and functions of hypergeometric type