RT Journal Article
T1 Δ-Sobolev orthogonal polynomials of Meixner type: asymptotics and limit relation
A1 Area, Iván
A1 Godoy, Eduardo
A1 Marcellán Español, Francisco José
A1 Moreno Balcázar, Juan José
AB Let $\{Q_n(x)\}_n$ be the sequence of monic polynomials orthogonal with respect to the Sobolev-type inner product $$\bigl\langle (p(x),r(x)\bigr \rangle_S=\bigl\langle{\bold u}_0,p(x)r(x) \bigr\rangle+ \lambda\bigl\langle {\bold u}_1,(\Delta p)(x)(\Delta r)(x) \bigr\rangle,$$ where $\lambda\ge 0$, $(\Delta f)(x)=f(x+1)-f(x)$ denotes the forward difference operator and $({\bold u}_0,{\bold u}_1)$ is a $\Delta$-coherent pair of positive-definite linear functionals being ${\bold u}_1$ 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 $\bbfC\setminus[0,+\infty)$ 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-$\Delta$-coherent pair, that is, when ${\bold u}_0={\bold u}_1$ 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.
PB Elsevier
SN 0377-0427
YR 2005
FD 2005-06
LK https://hdl.handle.net/10016/5953
UL https://hdl.handle.net/10016/5953
LA eng
NO 16 pages, no figures.-- MSC2000 codes: 42C05.-- Issue title: "Proceedings of the Seventh International Symposium on Orthogonal Polynomials, Special Functions and Applications" (University of Copenhagen, Denmark, Aug 18-22, 2003).
NO MR#: MR2127867 (2006a:33005)
NO Zbl#: Zbl 1060.42015
NO The work by I.A. and E.G. was partially supported by Ministerio de Ciencia y Tecnología of Spain under grant BFM2002-04314-C02-01. The work by F.M. has been supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grant BFM2003-06335-C03-02 as well as by the NATO collaborative grant PST.CLG. 979738. The work by J.J.M.B has been supported by Dirección General de Investigación of Spain under grant BFM2001-3878-C02-02 as well as by Junta de Andalucía (research group FQM0229).
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