Orthogonal matrix polynomials satisfying differential equations with recurrence coefficients having non-scalar limits

Borrego, J. T.; Castro, Mirta M.; Durán, Antonio J.

doi:10.1080/10652469.2011.627510

Cited by 14 publications

(26 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, where it was shown that the polynomials that satisfy a relation of the form

x P_{k} false(x false) = A_{k} P_{k + 1} false(x false) + B_{k} P_{k} false(x false) + A_{k - 1}^{*} P_{k - 1} false(x false)

for

k = 0, 1, \dots

, are orthogonal with respect to a positive definite matrix of measures; ie, a matrix version of Favard's theorem. Then, in the 1990s and the 2000s, it was found that MOPs satisfy in some cases similar properties as the classical orthogonal polynomials, ie, the scalar‐type Rodrigues' formula and a second‐order differential equation . It has been proven that operators of the form D =

\partial^{2} F_{2} false(t false)

\partial^{1} F_{1} false(t false)

\partial^{0} F_{0}

have eigenfunctions of different infinite families of MOPs.…”

Section: Introductionmentioning

confidence: 98%

“…A new family of MOPs satisfying second‐order differential equations whose coefficients do not behave asymptotically as the identity matrix was found in Ref. ; see also Ref. .…”

Section: Introductionmentioning

confidence: 99%

“…Then, in the 1990s and the 2000s, it was found that MOPs satisfy in some cases similar properties as the classical orthogonal polynomials, ie, the scalartype Rodrigues' formula 38,39 and a second-order differential equation. [40][41][42] It has been proven 43 that operators of the form = 2 2 ( ) + 1 1 ( ) + 0 0 have eigenfunctions of different infinite families of MOPs. A new family of MOPs satisfying second-order differential equations whose coefficients do not behave asymptotically as the identity matrix was found in Ref.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Riemann‐Hilbert problem and matrix discrete Painlevé II systems

Cassatella‐Contra

Mañas

2019

Stud Appl Math

View full text Add to dashboard Cite

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.

show abstract

“…, where it was shown that the polynomials that satisfy a relation of the form

x P_{k} false(x false) = A_{k} P_{k + 1} false(x false) + B_{k} P_{k} false(x false) + A_{k - 1}^{*} P_{k - 1} false(x false)

for

k = 0, 1, \dots

\partial^{2} F_{2} false(t false)

\partial^{1} F_{1} false(t false)

\partial^{0} F_{0}

have eigenfunctions of different infinite families of MOPs.…”

Section: Introductionmentioning

confidence: 98%

“…A new family of MOPs satisfying second‐order differential equations whose coefficients do not behave asymptotically as the identity matrix was found in Ref. ; see also Ref. .…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Riemann‐Hilbert problem and matrix discrete Painlevé II systems

Cassatella‐Contra

Mañas

2019

Stud Appl Math

View full text Add to dashboard Cite

show abstract

“…The study of such polynomials goes back at least to [19] and we refer to the survey paper [8] for a detailed discussion of the available literature and for many more references. Some recent developments and applications of matrix orthogonal polynomials can be found in [4,6,9,17] among many others.…”

Section: Introductionmentioning

confidence: 99%

Zeros and ratio asymptotics for matrix orthogonal polynomials

Delvaux

Dette

2012

JAMA

View full text Add to dashboard Cite

Ratio asymptotics for matrix orthogonal polynomials with recurrence coefficients A n and B n having limits A and B respectively (the matrix Nevai class) were obtained by Durán. In the present paper we obtain an alternative description of the limiting ratio. We generalize it to recurrence coefficients which are asymptotically periodic with higher periodicity, and/or which are slowly varying in function of a parameter. Under such assumptions, we also find the limiting zero distribution of the matrix orthogonal polynomials, generalizing results by Durán-López-Saff and Dette-Reuther to the non-Hermitian case. Our proofs are based on 'normal family' arguments and on the solution to a quadratic eigenvalue problem. As an application of our results we obtain new explicit formulas for the spectral measures of the matrix Chebyshev polynomials of the first and second kind, and we derive the asymptotic eigenvalue distribution for a class of random band matrices generalizing the tridiagonal matrices introduced by Dumitriu-Edelman.

show abstract

“…This is a matrix version of Favard's theorem for scalar orthogonal polynomials. Then, in the 1990's and the 2000's some authors found that matrix orthogonal polynomials (MOP) satisfy in certain cases some properties that satisfy scalar valued orthogonal polynomials; for example, Laguerre, Hermite and Jacobi polynomials, i.e., the scalar-type Rodrigues' formula [44,45,34] and a second order differential equation [46,47,24]. Later on, it has been proven [48] that operators of the form D=∂ 2 F 2 (t)+∂ 1 F 1 (t)+∂ 0 F 0 have as eigenfunctions different infinite families of MOP's.…”

mentioning

confidence: 99%

Matrix orthogonal Laurent polynomials on the unit circle and Toda type integrable systems

Ariznabarreta

Mañas

2014

Advances in Mathematics

View full text Add to dashboard Cite

Abstract. Matrix orthogonal Laurent polynomials in the unit circle and the theory of Toda-like integrable systems are connected using the Gauss-Borel factorization of two, left and a right, Cantero-Morales-Velázquez block moment matrices, which are constructed using a quasi-definite matrix measure. A block Gauss-Borel factorization problem of these moment matrices leads to two sets of biorthogonal matrix orthogonal Laurent polynomials and matrix Szegő polynomials, which can be expressed in terms of Schur complements of bordered truncations of the block moment matrix. The corresponding block extension of the Christoffel-Darboux theory is derived. Deformations of the quasidefinite matrix measure leading to integrable systems of Toda type are studied. The integrable theory is given in this matrix scenario; wave and adjoint wave functions, Lax and Zakharov-Shabat equations, bilinear equations and discrete flows -connected with Darboux transformations-. We generalize the integrable flows of the Cafasso's matrix extension of the Toeplitz lattice for the Verblunsky coefficients of Szegő polynomials. An analysis of the Miwa shifts allows for the finding of interesting connections between Christoffel-Darboux kernels and Miwa shifts of the matrix orthogonal Laurent polynomials.

show abstract

Orthogonal matrix polynomials satisfying differential equations with recurrence coefficients having non-scalar limits

Cited by 14 publications

References 21 publications

Riemann‐Hilbert problem and matrix discrete Painlevé II systems

Riemann‐Hilbert problem and matrix discrete Painlevé II systems

Zeros and ratio asymptotics for matrix orthogonal polynomials

Matrix orthogonal Laurent polynomials on the unit circle and Toda type integrable systems

Contact Info

Product

Resources

About