Inner products involving differences: the meixner—sobolev polynomials

Area, Iván; Godoy, E.; Marcellán, Francisco

doi:10.1080/10236190008808211

Cited by 15 publications

(21 citation statements)

References 19 publications

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“…The polynomials {S n } were introduced for the first time in [1], and they are the so-called Δ-Meixner-Sobolev orthogonal polynomials. They are normalized by the condition that the leading coefficient of S n (x) equals the leading coefficient of m n (x; β, c), n ≥ 0.…”

Section: δ-Meixner-sobolev Orthogonal Polynomials: the Known Factsmentioning

confidence: 99%

A generating function for nonstandard orthogonal polynomials involving differences: the Meixner case

2011

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In this paper we deal with a family of nonstandard polynomials orthogonal with respect to an inner product involving differences. This type of inner product is the so-called Δ-Sobolev inner product. Concretely, we consider the case in which both measures appearing in the inner product correspond to the Pascal distribution (the orthogonal polynomials associated to this distribution are known as Meixner polynomials). The aim of this work is to obtain a generating function for the Δ-Meixner-Sobolev orthogonal polynomials and, by using a limit process, recover a generating function for Laguerre-Sobolev orthogonal polynomials.

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Section: δ-Meixner-sobolev Orthogonal Polynomials: the Known Factsmentioning

confidence: 99%

A generating function for nonstandard orthogonal polynomials involving differences: the Meixner case

2011

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“…To obtain a uniform asymptotic approximation, we will use the cubic transformation suggested by Chester, Friedman and Ursell [4] f (t) = 1 3…”

Section: The Casementioning

confidence: 99%

“…When λ = 0 the Meixner-Sobolev polynomials reduce to the classical Meixner polynomials. These polynomials were introduced in [1], and a recurrence relation involving S n , S n−1 and 2 classical Meixner polynomials are given in [2] and [7]. This recurrence relation is very useful for generating the polynomials.…”

Section: Introductionmentioning

confidence: 99%

“…The graph of a rescaled version of S 30 (x), where a =5 6 , c =1 3 and β = 8 . Note the dramatic changes at x = nY − ≈ 8 and at x = nY + ≈ 112.…”

mentioning

confidence: 99%

“…The graph of a rescaled version of S 20 (x), where a =5 6 , c =1 3 , β = 9 8 (black), and approximation (6.18) (grey). Note that only near y = 0 and y = Y + the difference is visible.…”

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confidence: 99%

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Uniform Asymptotic Approximations for the Meixner–sobolev Polynomials

Khwaja

Daalhuis

2012

Anal. Appl.

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We obtain uniform asymptotic approximations for the monic Meixner–Sobolev polynomials Sn(x). These approximations for n → ∞, are uniformly valid for x/n restricted to certain intervals, and are in terms of Airy functions. We also give asymptotic approximations for the location of the zeros of Sn(x), especially the small and the large zeros are discussed. As a limit case, we also give a new asymptotic approximation for the large zeros of the classical Meixner polynomials. The method is based on an integral representation in which a hypergeometric function appears in the integrand. After a transformation, the hypergeometric functions can be uniformly approximated by unity, and all that remains are simple integrals for which standard asymptotic methods are used. As far as we are aware, this is the first time that standard uniform asymptotic methods have been used for the Sobolev-class of orthogonal polynomials.

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New analytic properties of nonstandard Sobolev-type Charlier orthogonal polynomials

Huertas

Soria-Lorente

2018

Numer Algor

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In this contribution we consider the sequence {Q λ n } n≥0 of monic polynomials orthogonal with respect to the following inner product involving differenceswhere λ ∈ R + , ∆ denotes the forward difference operator defined by ∆f (x) = f (x + 1) − f (x), ψ (a) with a > 0 is the well known Poisson distribution of probability theory dψ (a) (x) = e −a a x x! at x = 0, 1, 2, . . . , and c ∈ R is such that ψ (a) has no points of increase in the interval (c, c + 1). We derive its corresponding hypergeometric representation. The ladder operators and two different versions of the linear difference equation of second order corresponding to these polynomials are given. Recurrence formulas of five and three terms, the latter with rational coefficients, are presented. Moreover, for real values of c such that c + 1 < 0, we obtain some results on the distribution of its zeros as decreasing functions of λ, when this parameter goes from zero to infinity.

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Inner products involving differences: the meixner—sobolev polynomials

Cited by 15 publications

References 19 publications

A generating function for nonstandard orthogonal polynomials involving differences: the Meixner case

A generating function for nonstandard orthogonal polynomials involving differences: the Meixner case

Uniform Asymptotic Approximations for the Meixner–sobolev Polynomials

New analytic properties of nonstandard Sobolev-type Charlier orthogonal polynomials

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