On linearly related sequences of derivatives of orthogonal polynomials

• Autor(a) principal: Jesus, M. N. de
• Data de Publicação: 2008
• Outros Autores: Petronilho, J.
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10316/4582; https://doi.org/10.1016/j.jmaa.2008.06.017
• topic: Orthogonal polynomials; Inverse problems; Semiclassical orthogonal polynomials; Stieltjes transforms

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 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[less-than-or-equals, slant]m[less-than-or-equals, slant]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 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.