Burchnall Formulas and Non-Abelian Toda Lattice (original) (raw)

Matrix valued Hermite polynomials, Burchnall formulas and non-abelian Toda lattice

Advances in Applied Mathematics, 2019

A general family of matrix valued Hermite type orthogonal polynomials is introduced and studied in detail by deriving Pearson equations for the weight and matrix valued differential equations for these matrix polynomials. This is used to derive Rodrigues formulas, explicit formulas for the squared norm and to give an explicit expression of the matrix entries as well to derive a connection formula for the matrix polynomials of Hermite type. We derive matrix valued analogues of Burchnall formulas in operational form as well explicit expansions for the matrix valued Hermite type orthogonal polynomials as well as for previously introduced matrix valued Gegenbauer type orthogonal polynomials. The Burchnall approach gives two descriptions of the matrix valued orthogonal polynomials for the Toda modification of the matrix weight for the Hermite setting. In particular, we obtain a non-trivial solution to the non-abelian Toda lattice equations. − B (α,ν) n−1 consists of a product of diagonal matrices and A *. Since we have T as a diagonal matrix, the commutation shows that [T, A * ] = 0. This means that all diagonal elements of T are equal, and the result follows.

An algebraic theory about semiclassical and classical matrix orthogonal polynomials

In this paper we introduce an algebraic theory of classical matrix orthogonal polynomials as a particular case of the semi-classical ones, defined by a distributional equation for the corresponding orthogonality functional. This leads to several properties that characterize the classical matrix families, among them, a structure relation and a second order differo-differential equation. In the particular case of Hermite type matrix polynomials we obtain all the parameters associated with the family and we prove that they satisfy, not only a differo-differential equation, but a second order differential one, as it can be seen in the scalar case.

Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions

Symmetry, Integrability and Geometry: Methods and Applications, 2018

Burchnall's method to invert the Feldheim-Watson linearization formula for the Hermite polynomials is extended to all polynomial families in the Askey-scheme and its q-analogue. The resulting expansion formulas are made explicit for several families corresponding to measures with infinite support, including the Wilson and Askey-Wilson polynomials. An integrated version gives the possibility to give alternate expression for orthogonal polynomials with respect to a modified weight. This gives expansions for polynomials, such as Hermite, Laguerre, Meixner, Charlier, Meixner-Pollaczek and big q-Jacobi polynomials and big q-Laguerre polynomials. We show that one can find expansions for the orthogonal polynomials corresponding to the Toda-modification of the weight for the classical polynomials that correspond to known explicit solutions for the Toda lattice, i.e., for Hermite, Laguerre, Charlier, Meixner, Meixner-Pollaczek and Krawtchouk polynomials.

On Matrix polynomials associated with Hermite polynomials

Tamkang Journal of Mathematics, 2015

In this paper, an extension of the Hermite matrix polynomials is introduced. Some relevant matrix functions appear in terms of the two-index and two-variable and p-index and p-variable Hermite matrix polynomials. Furthermore, in order to give qualitative properties of this family of matrix polynomials, the Legendre and Chebyshev matrix polynomials of sveral variables are introduced.

Orthogonal matrix polynomials, scalar-type Rodrigues’ formulas and Pearson equations

Journal of Approximation Theory, 2005

Some families of orthogonal matrix polynomials satisfying second order differential equations with coefficients independent of n have recently been introduced (see ). An important difference with the scalar classical families of Jacobi, Laguerre and Hermite, is that these matrix families do not satisfy scalar type Rodrigues' formulas of the type (Φ n W ) (n) W −1 , where Φ is a matrix polynomial of degree not bigger than 2. An example of a modified Rodrigues' formula, well suited to the matrix case, appears in .

On generalized two-index Hermite matrix polynomials

Miskolc Mathematical Notes, 2017

In this study, we give a new generalization for Hermite matrix polynomials. We obtain some formulas related to an explicit representation, matrix recurrence relations and the Rodrigues formula for two-index Hermite matrix polynomials. Also we generalize Runge's addition formula and Nielsen's identity for these new generalizations. Moreover we give Burchnall's operational formula and Nielsen's identity for Hermite matrix polynomials by means of two-index Hermite matrix polynomials.