A Generalization of Laguerre Polynomials

Koekoek, Roelof; Meijer, H. G.

doi:10.1137/0524047

Cited by 80 publications

(56 citation statements)

References 6 publications

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“…, where the coefficients {β i (x)} ∞ i=0 are given by β 0 (0, α) = 0 and (11), (12), (13), (14), (15) and (16), has order 2α + 8 if α is a nonnegative integer and has infinite order otherwise.…”

Section: Differential Equations For Laguerre Polynomials 351mentioning

confidence: 99%

On differential equations for Sobolev-type Laguerre polynomials

Koekoek

Bavinck

1998

Trans. Amer. Math. Soc.

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Abstract. The Sobolev-type Laguerre polynomials {L α,M,N n (x)} ∞ n=0 are orthogonal with respect to the inner product,In 1990 the first and second author showed that in the case M > 0 and N = 0 the polynomials are eigenfunctions of a unique differential operator of the formwhereare independent of n. This differential operator is of order 2α + 4 if α is a nonnegative integer, and of infinite order otherwise.In this paper we construct all differential equations of the formwhere the coefficientsare independent of n and the coefficients a 0 (x), b 0 (x) and c 0 (x) are independent of x, satisfied by the Sobolev-type Laguerre polynomials {L α,M,N n (x)} ∞ n=0 . Further, we show that in the case M = 0 and N > 0 the polynomials are eigenfunctions of a linear differential operator, which is of order 2α + 8 if α is a nonnegative integer and of infinite order otherwise.Finally, we show that in the case M > 0 and N > 0 the polynomials are eigenfunctions of a linear differential operator, which is of order 4α + 10 if α is a nonnegative integer and of infinite order otherwise.

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Section: Differential Equations For Laguerre Polynomials 351mentioning

confidence: 99%

On differential equations for Sobolev-type Laguerre polynomials

Koekoek

Bavinck

1998

Trans. Amer. Math. Soc.

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“…Special attention has been paid to their algebraic and analytic properties of these polynomials, in particular, the distribution of their zeros taking into account the location of the point ξ with respect to the set E. When E is the interval [0, +∞) and ξ = 0, Meijer [1993a] analyzed some analytic properties of the zeros of the so called discrete Sobolev orthogonal polynomials (1). Some results of [Meijer 1993a] are direct generalizations of the results of [Koekoek and Meijer 1993], where the weight function is the Laguerre weight ω(x) = x α e −x . Koekoek and Meijer established properties of the discrete Laguerre-Sobolev polynomials such as their representation as a hypergeometric series, an holonomic second order linear differential equation associated with them, properties of the zeros, and a higher-order recurrence relation that such polynomials satisfy.…”

Section: Introductionmentioning

confidence: 98%

On orthogonal polynomials with respect to certain discrete Sobolev inner product

Marcellán¹,

Zejnullahu²,

Fejzullahu³

et al. 2012

Pacific J. Math.

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In this paper we deal with sequences of polynomials orthogonal with respect to the discrete Sobolev inner productwhere ω is a weight function, ξ ≤ 0, and M, N ≥ 0. The location of the zeros of discrete Sobolev orthogonal polynomials is given in terms of the zeros of standard polynomials orthogonal with respect to the weight function ω. In particular, for ω(x) = x α e −x we obtain the asymptotics for discrete Laguerre-Sobolev orthogonal polynomials.

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“…The zeros of the respective Sobolev orthogonal polynomials are studied by Meijer in [33]. [19] and R. Koekoek [18] and generalized to arbitrary measures d λ ac 0 on [0, ∞] by H.G. Meijer [31].…”

Section: Introductionmentioning

confidence: 99%

Computing orthogonal polynomials in Sobolev spaces

Gautschi

Zhang

1995

Numerische Mathematik

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Numerical methods are considered for generating polynomials orthogonal with respect to an inner product of Sobolev type, i.e., one that involves derivatives up to some given order, each having its own (positive) measure associated with it. The principal objective is to compute the coefficients in the increasing-order recurrence relation that these polynomials satisfy by virtue of them forming a sequence of monic polynomials with degrees increasing by 1 from one member to the next. As a by-product of this computation, one gains access to the zeros of these polynomials via eigenvalues of an upper Hessenberg matrix formed by the coefficients generated. Two methods are developed: One is based on the modified moments of the constitutive measures and generalizes what for ordinary orthogonal polynomials is known as "modified Chebyshev algorithm". The other -a generalization of "Stieltjes's procedure" -expresses the desired coefficients in terms of a Sobolev inner product involving the orthogonal polynomials in question, whereby the inner product is evaluated by numerical quadrature and the polynomials involved are computed by means of the recurrence relation already generated up to that point. The numerical characteristics of these methods are illustrated in the case of Sobolev orthogonal polynomials of old as well as new types. Based on extensive numerical experimentation, a number of conjectures are formulated with regard to the location and interlacing properties of the respective zeros. Mathematics Subject Classification (1991): 65D15, 42C05

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A Generalization of Laguerre Polynomials

Cited by 80 publications

References 6 publications

On differential equations for Sobolev-type Laguerre polynomials

On differential equations for Sobolev-type Laguerre polynomials

On orthogonal polynomials with respect to certain discrete Sobolev inner product

Computing orthogonal polynomials in Sobolev spaces

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