Riemann-Hilbert Problems for Multiple Orthogonal Polynomials

Assche, Walter Van; Geronimo, Jeffrey S.; Kuijlaars, Arno B. J.

doi:10.1007/978-94-010-0818-1_2

Cited by 100 publications

(98 citation statements)

References 18 publications

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“…Lemma 1.3 allows us to obtain the asymptotic properties of the polynomials P n,n through the analysis of a matrix-valued Riemann-Hilbert problem of size (d + 1) × (d + 1) that encodes the multiple orthogonality conditions (1.32). The first characterization of multiple orthogonality in terms of a RH problem appeared in [35]. The RH problem we will analyze is described in Section 4.…”

Section: Orthogonality On (D + 1)-star and Multiple Orthogonalitymentioning

confidence: 99%

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The normal matrix model with a monomial potential, a vector equilibrium problem, and multiple orthogonal polynomials on a star

Kuijlaars

López-García

2015

Nonlinearity

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We investigate the asymptotic behavior of a family of multiple orthogonal polynomials that is naturally linked with the normal matrix model with a monomial potential of arbitrary degree d + 1. The polynomials that we investigate are multiple orthogonal with respect to a system of d analytic weights defined on a symmetric (d + 1)-star centered at the origin. In the first part we analyze in detail a vector equilibrium problem involving a system of d interacting measures (µ1, . . . , µ d ) supported on star-like sets in the plane. We show that in the subcritical regime, the first component µ * 1 of the solution to this problem is the asymptotic zero distribution of the multiple orthogonal polynomials. It also characterizes the domain where the eigenvalues in the normal matrix model accumulate, in the sense that the Schwarz function associated with the boundary of this domain can be expressed explicitly in terms of µ * 1 . The second part of the paper is devoted to the asymptotic analysis of the multiple orthogonal polynomials. The asymptotic results are obtained again in the subcritical regime, and they follow from the Deift/Zhou steepest descent analysis of a Riemann-Hilbert problem of size (d + 1) × (d + 1). The vector equilibrium problem and the Riemann-Hilbert problem that we investigate are generalizations of those studied recently by Bleher-Kuijlaars in the case d = 2.

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Section: Orthogonality On (D + 1)-star and Multiple Orthogonalitymentioning

confidence: 99%

“…This is based on the multiple orthogonality stated in Lemma 1.3. As a result [35] there is a Riemann Hilbert problem of size (d + 1) × (d + 1).…”

Section: The Riemann-hilbert Problemmentioning

confidence: 99%

The normal matrix model with a monomial potential, a vector equilibrium problem, and multiple orthogonal polynomials on a star

Kuijlaars

López-García

2015

Nonlinearity

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“…The Riemann-Hilbert (RH) problem of Fokas-lts-Kitaev for orthogonal polynomials was generahzed by Van Assche, Geronimo and Kuijlaars [6] to characterize multiple orthogonal polynomials. Our starting point is this RH problem, and more specifically, its fundamental solution /, which is a « x « matrix such that the multiple orthogonal polynomials are elements in the first row.…”

Section: Z" = Fdmexp(-tr (-M^ -Am))mentioning

confidence: 99%

“…Again we assume that the solution P(n,x) of (8) is unique for each n G N?. The RH problem which characterizes these polynomials [5], [6] are determined by for type I and type II orthogonal polynomials respectively. Here 0 stands by the vector of C?…”

Section: Riemann-hilbert Problems and Multiple Orthogonal Polynomialsmentioning

confidence: 99%

Analysis of the Large-n Limit of Multiple Orthogonal Polynomials and Non-intersecting Brownian Motions

Alonso

Медина

2009

AIP Conference Proceedings

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Abstract. In this work we use Riemann-Hilbert problems for multiple orthogonal polynomials in order to derive string equations associated to Lax-Orlov pairs operators. These string equations provide us with a useful tool to analyze the large n-limit of the related hierarchies. The results are finally applied to the study of the associated random matrix models (Gaussian Hermitian matrix models with an external source) and non-intersecting Brownian motions starting from a fix point.

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“…Furthermore, Kuijlaars et al [12,13] considered the asymptotics of the polynomials that are orthogonal with respect to the modified Jacobi weight . For other investigations of a similar nature, we refer to [3,4,6,10,18]. An advantage of this new approach over the more classical asymptotic methods is that it is applicable to orthogonal polynomials which do not have an integral representation or satisfy a second-order differential equation.…”

Section: Introductionmentioning

confidence: 99%

Linear difference equations with transition points

Wang

Wong

2004

Math. Comp.

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Abstract. Two linearly independent asymptotic solutions are constructed for the second-order linear difference equationwhere An and Bn have power series expansions of the formβs n s with α 0 = 0. Our results hold uniformly for x in an infinite interval containing the transition point x + given by α 0 x + + β 0 = 2. As an illustration, we present an asymptotic expansion for the monic polynomials πn(x) which are orthogonal with respect to the modified Jacobi weight w(

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Riemann-Hilbert Problems for Multiple Orthogonal Polynomials

Cited by 100 publications

References 18 publications

The normal matrix model with a monomial potential, a vector equilibrium problem, and multiple orthogonal polynomials on a star

The normal matrix model with a monomial potential, a vector equilibrium problem, and multiple orthogonal polynomials on a star

Analysis of the Large-n Limit of Multiple Orthogonal Polynomials and Non-intersecting Brownian Motions

Linear difference equations with transition points

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