On a new class of 2-orthogonal polynomials, I: the recurrence relations and some properties

Douak, Khalfa; Maroni, P.

doi:10.1080/10652469.2020.1811702

Cited by 6 publications

(3 citation statements)

References 21 publications

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“…As (𝑥) satisfies Equation (39) and on account of the degrees of the polynomial entries in the matrices Φ(𝑥) and Ψ(𝑥), then Proposition 2.6 in Ref. 8 ensures the 2-orthogonality of the polynomial sequence ((𝑛 + 1) −1 𝑃 ′ 𝑛+1 (𝑥; 𝑎, 𝑏; 𝑐, 𝑑)) 𝑛∈ℕ with respect to the vector of weights 𝑥Φ(𝑥)(𝑥) whilst Proposition 2.7 in Ref.…”

Section: Differential Propertiesmentioning

confidence: 99%

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: Differential Propertiesmentioning

confidence: 99%

Multiple orthogonal polynomials with respect to Gauss' hypergeometric function

Lima

Loureiro

2021

Stud Appl Math

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“…The 2-orthogonal polynomials investigated in [22] satisfy orthogonality conditions with respect to weight functions V (x; a, b; c) and V (x; a, b; c+1), supported in R + , with a, b, c ∈ R + such that c > max{a, b} and where U (α, β ; x) is the confluent hypergeometric function of the second kind, also known as the Tricomi function (see [8, §13] for the definition and some properties of the confluent hypergeometric functions). These 2-orthogonal polynomials have also appeared in [10]. As shown in [22,Th.…”

Section: Connection With Other Hahn-classical 2-orthogonal Polynomialsmentioning

confidence: 77%

Multiple orthogonal polynomials with respect to Gauss' hypergeometric function

Lima¹,

Loureiro²

2020

Preprint

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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 have direct applications in the investigation of singular values of products of Ginibre matrices, in the analysis of rational solutions to Painlevé equations 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 indexes lie on the step line, for which it is shown that a differentiation on the variable gives a shift on the parameters, therefore satisfying the Hahn's property. We obtain a Rodrigues-type formula for the type I, while a more detailed characterisation is given for the type II polynomials (aka 2-orthogonal polynomials) which includes: an explicit expression as a terminating hypergeometric series, a third-order differential equation and a third-order recurrence relation. The asymptotic behaviour of their recurrence coefficients mimics those of Jacobi-Piñeiro polynomials, based on which, their zero asymptotic distribution and a Mehler-Heine asymptotic formula near the origin are given. Particular choices on the parameters degenerate in some known systems such as special cases of the Jacobi-Piñeiro polynomials, Jacobi-type 2-orthogonal polynomials and components of the cubic decomposition of threefold symmetric Hahn-classical polynomials. Equally considered are confluence relations to other known polynomial sets, such as multiple orthogonal polynomials with respect to Tricomi functions.

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“…. , u d−1 ) T , we refer to the comments provided in [9] (Remarks 1.2) which shed some light on the subject.…”

Section: Other Functional Identitiesmentioning

confidence: 99%

On a General Family of 2-Orthogonal Polynomial Eigenfunctions of a Third Order Differential Equation via Symbolic Computation

Mesquita

2024

Math.Comput.Sci.

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The search for 2-orthogonal polynomial eigenfunctions, with respect to a third order differential operator that does not increase the degree of polynomials, was recently developed in [23] by means of a symbolic approach. This work allowed us to establish some impossible cases as also to present a few families of such 2-orthogonal polynomial sequences. In this paper, we apply the symbolic setup proposed in [23] in order to enlighten us about further 2-orthogonal polynomial solutions of this problem. Concerning a general family inhere described, it is also proved its Hahn-classical character. Additionally, some functional identities are established.

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On a new class of 2-orthogonal polynomials, I: the recurrence relations and some properties

Cited by 6 publications

References 21 publications

Multiple orthogonal polynomials with respect to Gauss' hypergeometric function

Multiple orthogonal polynomials with respect to Gauss' hypergeometric function

Multiple orthogonal polynomials with respect to Gauss' hypergeometric function

On a General Family of 2-Orthogonal Polynomial Eigenfunctions of a Third Order Differential Equation via Symbolic Computation

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