An Introduction to Multiple Orthogonal Polynomials and Hermite-Padé Approximation

Lagomasino, G. López

doi:10.1007/978-3-030-56190-1_9

Cited by 7 publications

(6 citation statements)

References 31 publications

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“…To prove this result, we use the connection between continued fractions and Stieltjes transforms to guarantee the existence of an integral representation of the type in (13) for the ratio of the weight functions involved. The continued fractions relevant to this work are the so-called Stieltjes continued fractions or, simply, S-fractions, whose connection with Stieltjes transforms was first investigated in Ref.…”

Section: Nikishin Systemmentioning

confidence: 99%

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Multiple orthogonal polynomials with respect to Gauss' hypergeometric function

Lima

Loureiro

2021

Stud Appl Math

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A new set of multiple orthogonal polynomials of both type I and type II with respect to two weight functions involving Gauss' hypergeometric function on the interval (0,1) is studied. This type of polynomials has direct applications in the investigation of singular values of products of Ginibre random matrices and are connected with branched continued fractions and total-positivity problems in combinatorics. The pair of orthogonality measures is shown to be a Nikishin system and to satisfy a matrix Pearson-type differential equation. The focus is on the polynomials whose indices lie on the step-line, for which it is shown that the differentiation gives a shift in the parameters, therefore satisfying Hahn's property. We obtain Rodrigues-type formulas for type I polynomials and functions, while a more detailed characterization is given for the type II polynomials (aka 2-orthogonal polynomials) that include an explicit expression as a terminating hypergeometric series, a third-order differential equation, and a third-order recurrence relation. The asymptotic behavior of their recurrence coefficients mimics those of Jacobi-Piñeiro polynomials, based on which their asymptotic zero distribution and a Mehler-Heine asymptotic formula near the origin are given. Particular choices of the parameters and confluence relations give some known

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Section: Nikishin Systemmentioning

confidence: 99%

“…We give a brief introduction to this topic here, further information can be found, for instance, in Refs 12,Ch. 23] and [ 13 . When referring to (𝑃 𝑛 (𝑥)) 𝑛∈ℕ as a polynomial sequence, it is assumed that 𝑃 𝑛 is a polynomial of a single variable with degree exactly 𝑛.…”

Section: Introductionmentioning

confidence: 99%

Multiple orthogonal polynomials with respect to Gauss' hypergeometric function

Lima

Loureiro

2021

Stud Appl Math

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“…For further information about multiple orthogonal polynomials and Nikishin systems, we refer to [14,Ch. 23] and [19].…”

Section: H Lima and A F Loureiromentioning

confidence: 99%

Multiple orthogonal polynomials associated with confluent hypergeometric functions

Lima

Loureiro

2020

Journal of Approximation Theory

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“…Subsequently, they have been used in other related problems of number theory. In recent years, Hermite-Padé type approximations have received increasing attention because of their appearance in other areas such as the theories of multiple orthogonal polynomials, random matrix, non-intersecting Brownian motions [11,22,34,35] and as well as peakon problems [29,[37][38][39]. For more recent developments on other related topics, one might refer [3,4,36] etc.…”

mentioning

confidence: 99%

“…Moreover, it is noted that approximations of Nikishin systems (see e.g. [35][36][37] for more details on such systems of functions) are involved in these peakon problems.…”

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confidence: 99%

Hermite--Padé approximations with Pfaffian structures: Novikov peakon equation and integrable lattices

Chang¹

2021

Preprint

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Motivated by the Novikov equation and its peakon problem, we propose a new mixed type Hermite-Padé approximation whose unique solution is a sequence of polynomials constructed with the help of Pfaffians. These polynomials belong to the family of recently proposed partial-skew-orthogonal polynomials. The relevance of partialskew-orthogonal polynomials is especially visible in the approximation problem germane to the Novikov peakon problem so that we obtain explicit inverse formulae in terms of Pfaffians by reformulating the inverse spectral problem for the Novikov multipeakons.Furthermore, we investigate two Hermite-Padé approximations for the related spectral problem of the discrete dual cubic string, and show that these approximation problems can also be solved in terms of partial-skew-orthogonal polynomials and nonsymmetric Cauchy biorthogonal polynomials. This formulation results in a new correspondence among several integrable lattices.

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An Introduction to Multiple Orthogonal Polynomials and Hermite-Padé Approximation

Cited by 7 publications

References 31 publications

Multiple orthogonal polynomials with respect to Gauss' hypergeometric function

Multiple orthogonal polynomials with respect to Gauss' hypergeometric function

Multiple orthogonal polynomials associated with confluent hypergeometric functions

Hermite--Padé approximations with Pfaffian structures: Novikov peakon equation and integrable lattices

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