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Title: On linearly related sequences of derivatives of orthogonal polynomials
Authors: Jesus, M. N. de 
Petronilho, J. 
Keywords: Orthogonal polynomials; Inverse problems; Semiclassical orthogonal polynomials; Stieltjes transforms
Issue Date: 1-Sep-2008
Citation: Journal of Mathematical Analysis and Applications. In Press, Corrected Proof:
Abstract: 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.
Rights: openAccess
Appears in Collections:FCTUC Matemática - Artigos em Revistas Internacionais

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