Gaussian quadrature for multiple orthogonal polynomials

Coussement, J.; Assche, Walter Van

doi:10.1016/j.cam.2004.04.016

Cited by 39 publications

(50 citation statements)

References 17 publications

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“…The proof is similar to the proof of the three-terms recurrence relation satisfied by a sequence of orthogonal polynomials, see for instance [3]. Because of this recurrence relation, formal multiple orthogonal polynomials are a useful tool in the spectral theory of non-symmetric linear difference operators [14].…”

Section: Introductionmentioning

confidence: 87%

Multiple Wilson and Jacobi–Piñeiro polynomials

Beckermann

Coussement

Assche

2005

Journal of Approximation Theory

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We introduce multiple Wilson polynomials, which give a new example of multiple orthogonal polynomials (Hermite-Padé polynomials) of type II. These polynomials can be written as a Jacobi-Piñeiro transform, which is a generalization of the Jacobi transform for Wilson polynomials, found by Koornwinder. Here we need to introduce Jacobi and Jacobi-Piñeiro polynomials with complex parameters. Some explicit formulas are provided for both Jacobi-Piñeiro and multiple Wilson polynomials, one of them in terms of Kampé de Fériet series. Finally, we look at some limiting relations and construct a part of a multiple AT-Askey table.

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Section: Introductionmentioning

confidence: 87%

Multiple Wilson and Jacobi–Piñeiro polynomials

Beckermann

Coussement

Assche

2005

Journal of Approximation Theory

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“…This is an algebraic equation of order 4 and hence it has four solutions S (1) , S (2) , S (3) , S (4) . A careful analysis of these solutions and equation (3.2) near infinity shows that for z → ∞…”

Section: Zerosmentioning

confidence: 99%

Multiple Hermite polynomials and simultaneous Gaussian quadrature

Assche¹,

Vuerinckx²

2019

etna

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Multiple Hermite polynomials are an extension of the classical Hermite polynomials for which orthogonality conditions are imposed with respect to r > 1 normal (Gaussian) weights w j (x) = e −x 2 +c j x with different means c j /2, 1 ≤ j ≤ r. These polynomials have a number of properties, such as a Rodrigues formula, recurrence relations (connecting polynomials with nearest neighbor multi-indices), a differential equation, etc. The asymptotic distribution of the (scaled) zeros is investigated and an interesting new feature happens: depending on the distance between the c j , 1 ≤ j ≤ r, the zeros may accumulate on s disjoint intervals, where 1 ≤ s ≤ r. We will use the zeros of these multiple Hermite polynomials to approximate integrals of the form ∞ −∞ f (x) exp(−x 2 + c j x) dx simultaneously for 1 ≤ j ≤ r for the case r = 3 and the situation when the zeros accumulate on three disjoint intervals. We also give some properties of the corresponding quadrature weights. * Work supported EOS project 30889451 and FWO project G.0864.16N

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“…since the first of these integrals vanishes for k ≤ ⌊ n−1 2 ⌋ and ⌊ n 2 ⌋ − 1 ≤ ⌊ n−1 2 ⌋, see [6,Remark 2.2]. This degree of freedom will be reflected when we want to compute the recurrence coefficients of the orthogonal polynomials p n (x; µ 1 ) and p n (x; µ 2 ).…”

Section: The Recurrence Coefficients Along the Step-linementioning

confidence: 99%

Computing recurrence coefficients of multiple orthogonal polynomials

2015

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Multiple orthogonal polynomials satisfy a number of recurrence relations, in particular there is a (r + 2)-term recurrence relation connecting the type II multiple orthogonal polynomials near the diagonal (the so-called step-line recurrence relation) and there is a system of r recurrence relations connecting the nearest neighbors (the so-called nearest neighbor recurrence relations). In this paper we deal with two problems. First we show how one can obtain the nearest neighbor recurrence coefficients (and in particular the recurrence coefficients of the orthogonal polynomials for each of the defining measures) from the step-line recurrence coefficients. Secondly we show how one can compute the step-line recurrence coefficients from the recurrence coefficients of the orthogonal polynomials of each of the measures defining the multiple orthogonality.

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Gaussian quadrature for multiple orthogonal polynomials

Cited by 39 publications

References 17 publications

Multiple Wilson and Jacobi–Piñeiro polynomials

Multiple Wilson and Jacobi–Piñeiro polynomials

Multiple Hermite polynomials and simultaneous Gaussian quadrature

Computing recurrence coefficients of multiple orthogonal polynomials

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