Riemann–Hilbert Problems, Matrix Orthogonal Polynomials and Discrete Matrix Equations with Singularity Confinement

Cassatella‐Contra, Giovanni A.; Mañas, Manuel

doi:10.1111/j.1467-9590.2011.00541.x

Cited by 28 publications

(41 citation statements)

References 26 publications

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“…In a paper [11] of Chen and Zhang, the determinant of the Hankel matrix of the moments associated with the Jacobi weight perturbed by multiplying a Heaviside step function, is shown related to the sixth Painlevé equation PVI. More recently, in a paper [10] by Cassatella-Contra and Mañas, certain matrix orthogonal polynomials are described in terms of a Riemann-Hilbert problem. The corresponding discrete matrix recursion equation is derived, and it is noted that this equation might be considered as the matrix discrete Painlevé I equation Also, in [27], Xu and Zhao have used a Painlevé 34 transcendent to derive the asymptotics of the polynomials orthogonal with respect to the Gaussian weight with jump discontinuity at the edge of the equilibrium measure.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Asymptotics of Discrete Painlevé V transcendents via the Riemann–Hilbert Approach

Zhao

2013

Stud Appl Math

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We study a system of discrete Painlevé V equations via the Riemann-Hilbert approach. We begin with an isomonodromy problem for dPV, which admits a discrete Riemann-Hilbert problem formulation. The asymptotics of the discrete Riemann-Hilbert problem is derived via the nonlinear steepest descent method of Deift and Zhou. In the analysis, a parametrix is constructed in terms of specific Painlevé V transcendents. As a result, the asymptotics of the dPV transcendents are represented in terms of the PV transcendents. In the special case, our result confirms a conjecture of Borodin, that the difference Schlesinger equations converge to the differential Schlesinger equations at the solution level.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Asymptotics of Discrete Painlevé V transcendents via the Riemann–Hilbert Approach

Zhao

2013

Stud Appl Math

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“…This intrinsic fact, leagued with the multivariate character, lead in the one hand to the appearance of Schur complements and quasi-determinants and, on the other hand, to multivariate Cauchy integrals and integrals along the Shilov border of poly-disks -that is, to be faced to some basic facts of complex analysis in several variables. The Schur complement already appeared in the study of matrix orthogonal polynomials, see for example [14,26,27], but we did not understood yet in [14] the important role played int the theory by quasi-determinants; now we do. We adjacently get across symmetric algebra [43,75], being isomorphic to the set of multivariate polynomials it some times allow for simple derivation of some result or illuminate some structure.…”

Section: 2mentioning

confidence: 99%

Multivariate orthogonal polynomials and integrable systems

Ariznabarreta

Mañas

2016

Advances in Mathematics

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Abstract. Multivariate orthogonal polynomials in D real dimensions are considered from the perspective of the Cholesky factorization of a moment matrix. The approach allows for the construction of corresponding multivariate orthogonal polynomials, associated second kind functions, Jacobi type matrices and associated three term relations and also Christoffel-Darboux formulae. The multivariate orthogonal polynomials, its second kind functions and the corresponding Christoffel-Darboux kernels are shown to be quasi-determinants -as well as Schur complements-of bordered truncations of the moment matrix; quasi-tau functions are introduced. It is proven that the second kind functions are multivariate Cauchy transforms of the multivariate orthogonal polynomials. Discrete and continuous deformations of the measure lead to Toda type integrable hierarchy, being the corresponding flows described through Lax and Zakharov-Shabat equations; bilinear equations are found. Varying size matrix nonlinear partial difference and differential equations of the 2D Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials. The discrete flows, which are shown to be connected with a Gauss-Borel factorization of the Jacobi type matrices and its quasideterminants, lead to expressions for the multivariate orthogonal polynomials and its second kind functions in terms of shifted quasi-tau matrices, which generalize to the multidimensional realm those that relate the Baker and adjoint Baker functions with ratios of Miwa shifted τ -functions in the 1D scenario. In this context, the multivariate extension of the elementary Darboux transformation is given in terms of quasi-determinants of matrices built up by the evaluation, at a poised set of nodes lying in an appropriate hyperplane in R D , of the multivariate orthogonal polynomials. The multivariate Christoffel formula for the iteration of m elementary Darboux transformations is given as a quasi-determinant. It is shown, using congruences in the space of semi-infinite matrices, that the discrete and continuous flows are intimately connected and determine nonlinear partial difference-differential equations that involves only one site in the integrable lattice behaving as a Kadomstev-Petviashvili type system. Finally, a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials are given. In particular, it is shown that the Toda times that preserve the invariance condition lay in a secant variety of the Veronese variety of the fixed point set of the linear isometry.

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“…With the notation [0] = ( ), , , , ∈ ℂ × , a entrywise form of the system of matrix equations (52) and (53) is…”

Section: Proposition 22mentioning

confidence: 99%

“…In Ref. 52, the RHp for this matrix situation and the appearance of non-Abelian version of dPI was explored, showing singularity confinement. 53 The singularity analysis for a matrix version of dPI equation was performed.…”

Section: Introductionmentioning

confidence: 99%

Riemann‐Hilbert problem and matrix discrete Painlevé II systems

Cassatella‐Contra

Mañas

2019

Stud Appl Math

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Matrix Szegő biorthogonal polynomials for quasi‐definite matrices of Hölder continuous weights are studied. A Riemann‐Hilbert problem is uniquely solved in terms of the matrix Szegő polynomials and its Cauchy transforms. The Riemann‐Hilbert problem is given as an appropriate framework for the discussion of the Szegő matrix and the associated Szegő recursion relations for the matrix orthogonal polynomials and its Cauchy transforms. Pearson‐type differential systems characterizing the matrix of weights are studied. These are linear systems of ordinary differential equations that are required to have trivial monodromy. Linear ordinary differential equations for the matrix Szegő polynomials and its Cauchy transforms are derived. It is shown how these Pearson systems lead to nonlinear difference equations for the Verblunsky matrices and two examples, of Fuchsian and non‐Fuchsian type, are considered. For both cases, a new matrix version of the discrete Painlevé II equation for the Verblunsky matrices is found. Reductions of these matrix discrete Painlevé II systems presenting locality are discussed.

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Riemann–Hilbert Problems, Matrix Orthogonal Polynomials and Discrete Matrix Equations with Singularity Confinement

Cited by 28 publications

References 26 publications

Asymptotics of Discrete Painlevé V transcendents via the Riemann–Hilbert Approach

Asymptotics of Discrete Painlevé V transcendents via the Riemann–Hilbert Approach

Multivariate orthogonal polynomials and integrable systems

Riemann‐Hilbert problem and matrix discrete Painlevé II systems

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