Differential equations and Sobolev orthogonality

Jung, In‐Ha; Kwon, Kil Hyun; Lee, D. W.; Littlejohn, Lance L.

doi:10.1016/0377-0427(95)00111-5

Cited by 5 publications

(6 citation statements)

References 7 publications

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“…Koekoek, R. Koekoek and H. Bavinck [15]. Using a different approach, P. Iliev [9] has recently extended these results for a Laguerre-Sobolev inner product of the form p, q = (For other related papers see [10] and [11]).…”

Section: Introduction and Resultsmentioning

confidence: 99%

Differential equations for discrete Laguerre–Sobolev orthogonal polynomials

Durán

Iglesia

2015

Journal of Approximation Theory

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The aim of this paper is to study differential properties of orthogonal polynomials with respect to a discrete Laguerre-Sobolev bilinear form with mass point at zero. In particular we construct the orthogonal polynomials using certain Casorati determinants. Using this construction, we prove that they are eigenfunctions of a differential operator (which will be explicitly constructed). Moreover, the order of this differential operator is explicitly computed in terms of the matrix which defines the discrete Laguerre-Sobolev bilinear form. * Partially supported by MTM2012-36732-C03-03 (Ministerio de Economía y Competitividad), FQM-262, FQM-4643, FQM-7276 (Junta de Andalucía) and Feder Funds (European Union). The second author is also grateful to FOS studio in Panamá, for letting him use their facilities during the summer of 2013.

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Section: Introduction and Resultsmentioning

confidence: 99%

Differential equations for discrete Laguerre–Sobolev orthogonal polynomials

Durán

Iglesia

2015

Journal of Approximation Theory

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“…For more results on Sobolev orthogonality and spectral differential equations the reader is referred to [6]. Some results concerning the symmetrizability of the differential equations obtained in this paper can be found in [7].…”

Section: Introductionmentioning

confidence: 99%

“…Some results concerning the symmetrizability of the differential equations obtained in this paper can be found in [7].…”

Section: Satisfy a Differential Equation Of Formal Order 4α + 10 Whicmentioning

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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“…In 2003, discrete Jacobi-Sobolev orthogonal polynomials which are also common eigenfunctions of a higher-order differential operator entered into the picture. H. Bavinck (see [2]) proved that orthogonal polynomials with respect to the discrete Jacobi-Sobolev inner product (1), are eigenfunctions of a differential operator of infinite order, except for the following cases, where the order is finite and equals: For other related papers see [17,18,25].…”

Section: Rmentioning

confidence: 99%

“…Consider the sets U j , j = 1, 2, given by (A. 18). By construction (see (3.14)), we have that for h ∈ U j , the D-operator D h is defined by the sequence (ε n,j ) n (see (3.2) and (3.3)).…”

Section: Appendixmentioning

confidence: 99%

Differential equations for discrete Jacobi–Sobolev orthogonal polynomials

Durán

Iglesia

2018

J. Spectr. Theory

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The aim of this paper is to study differential properties of orthogonal polynomials with respect to a discrete Jacobi-Sobolev bilinear form with mass point at −1 and/or +1. In particular, we construct the orthogonal polynomials using certain Casorati determinants. Using this construction, we prove that when the Jacobi parameters α and β are nonnegative integers the Jacobi-Sobolev orthogonal polynomials are eigenfunctions of a differential operator of finite order (which will be explicitly constructed). Moreover, the order of this differential operator is explicitly computed in terms of the matrices which define the discrete Jacobi-Sobolev bilinear form.

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Differential equations and Sobolev orthogonality

Cited by 5 publications

References 7 publications

Differential equations for discrete Laguerre–Sobolev orthogonal polynomials

Differential equations for discrete Laguerre–Sobolev orthogonal polynomials

On differential equations for Sobolev-type Laguerre polynomials

Differential equations for discrete Jacobi–Sobolev orthogonal polynomials

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