The Genetic Sums' Representation for the Moments of a System of Stieltjes Functions and its Application

Aptekarev, A. I.; Kaliaguine, V.; Iseghem, Jeannette Van

doi:10.1007/s003650010004

Cited by 74 publications

(150 citation statements)

References 20 publications

Supporting

Mentioning

141

Contrasting

Unclassified

Order By: Relevance

“…Some references on this subject are [1,2,10,26,31]. See also [13,18] for the related study of Faber polynomials associated with hypocycloidal domains and stars.…”

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

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

View full text Add to dashboard Cite

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.

show abstract

“…Some references on this subject are [1,2,10,26,31]. See also [13,18] for the related study of Faber polynomials associated with hypocycloidal domains and stars.…”

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

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

View full text Add to dashboard Cite

show abstract

“…В конце прошлого века наблюдалась высокая активность в изучении вполне интегрируемых дискретных бесконечномерных динамических си-стем методом прямой-обратной спектральной задачи [1], в частности, це-почек Богоявленского [2], образующих иерархию над классическими це-почками Ленгмюра [3], [4], [5] и связанных с векторными непрерывными дробями Стилтьеса [6]. Термин вполне интегрируемые применительно к бесконечномерным системам, конечно, имеет некоторое условное зна-чение.…”

Section: Introductionunclassified

Weak asymptotics of multiple Pollaczek polynomials

Сорокин¹

2017

KIAM Prepr.

View full text Add to dashboard Cite

“…For example, in [1] and [2], Bogoyavlenskii gave a classification of these dynamical systems which are a discrete generalization of a KdV equation and showed that such systems have interesting applications on Hamilton mechanics. On the other hand in the work [3] a particular case of a Bogoyavlenskii discrete dynamical system appears related with the study of spectral problems for higher order difference equations and it was studied using a method based on the analysis of the genetic sums formula for the moments of the associated operator. Also in [4] the authors studied another particular case of these dynamical systems investigating the spectral properties of the associated band operator.…”

Section: Introductionmentioning

confidence: 99%

Dynamics and interpretation of some integrable systems via matrix orthogonal polynomials

Branquinho

Foulquié-Moreno

Mendes

2016

Integral Transforms and Special Functions

View full text Add to dashboard Cite

Abstract. In this work we characterize a high order Toda lattice in terms of a family of matrix polynomials orthogonal with respect to a complex matrix measure. In order to study the solution of this dynamical system we give explicit expressions for the Weyl function, generalized Markov function, and we also obtain, under some conditions, a representation of the vector of linear functionals associated with this system. We show that the orthogonality is embedded in these structure and governs the high order Toda lattice. We also present a Lax type theorem for the point spectrum of the Jacobi operator associated with a Toda type lattice.

show abstract

The Genetic Sums' Representation for the Moments of a System of Stieltjes Functions and its Application

Cited by 74 publications

References 20 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

Weak asymptotics of multiple Pollaczek polynomials

Dynamics and interpretation of some integrable systems via matrix orthogonal polynomials

Contact Info

Product

Resources

About