Δ-Sobolev orthogonal polynomials of Meixner type: asymptotics and limit relation

Authors:
I. Area;E. Godoy;F. Marcellán;J. J. Moreno-Balcázar
Affiliations:
Departamento de Matemática Aplicada II, E.T.S.E. de Telecomunicación, Universidade de Vigo,Campus Lagoas-Marcosende, 36200 Vigo, Spain;Departamento de Matemática Aplicada II, E.T.S.I. Industriales, Universidade de Vigo, Lagoas-Marcosende, 36200 Vigo, Spain;Departamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III, Avenida de la Universidad, 30, 28911 Leganés-Madrid, Spain;Departamento de Estadística y Matemática Aplicada, Edificio Científico Técnico III, Universidad de Almería, 04120 Almería, Spain, and Instituto Carlos I de Físic ...
Venue:
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the seventh international symposium on Orthogonal polynomials, special functions and applications
Year:
2005

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Abstract

Let {Q"n(x)}"n be the sequence of monic polynomials orthogonal with respect to the Sobolev-type inner product"S=+@l,where @l=0, (@Df)(x)=f(x+1)-f(x) denotes the forward difference operator and (u"0,u"1) is a @D-coherent pair of positive-definite linear functionals being u"1 the Meixner linear functional. In this paper, relative asymptotics for the {Q"n(x)}"n sequence with respect to Meixner polynomials on compact subsets of C@?[0,+~) is obtained. This relative asymptotics is also given for the scaled polynomials. In both cases, we deduce the same asymptotics as we have for the self-@D-coherent pair, that is, when u"0=u"1 is the Meixner linear functional. Furthermore, we establish a limit relation between these orthogonal polynomials and the Laguerre-Sobolev orthogonal polynomials which is analogous to the one existing between Meixner and Laguerre polynomials in the Askey scheme.