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Ratio and Plancherel-Rotach asymptotics for Meixner-Sobolev orthogonal polynomials

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2000-04-01
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Elsevier
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Abstract
We study the analytic properties of the monic Meixner-Sobolev polynomials $\{Q_n\}$ orthogonal with respect to the inner product involving differences $$(p,q)_S=\sum infty_{i=0}[p(i)q(i)+\lambda\Delta p(i)\Delta q(i)] {\mu (\gamma)_i\over i!},$$ $\gamma>0,\ 0<\mu<1$, where $\lambda\geq0,\ \Delta$ is the forward difference operator $(\Delta f(x)=f(x+1)-f(x))$ and $(\gamma)_n$ denotes the Pochhammer symbol. Relative asymptotics for Meixner-Sobolev polynomials with respect to Meixner polynomials is obtained. This relative asymptotics is also given for the scaled polynomials. Moreover, a zero distribution for the scaled Meixner-Sobolev polynomials and Plancherel-Rotach asymptotics for $\{Q_n\}$ are deduced.
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13 pages, no figures.-- MSC codes: 42C05; 33C25; 39A10.
MR#: MR1741786 (2000k:42032)
Zbl#: Zbl 0984.42016
Keywords
Sobolev orthogonal polynomials, Meixner polynomials, Asymptotics, Plancherel-Rotach asymptotics, Scaled polynomials, Zeroes
Bibliographic citation
Journal of Computational and Applied Mathematics, 2000, vol. 116, n. 1, p. 63-75