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Non Abelian Toda-type equations and matrix valued orthogonal polynomials

Alfredo Deaño, Lucía Morey, Pablo Román

Abstract

In this paper, we study parameter deformations of matrix valued orthogonal polynomials (MVOPs). These deformations are built on the use of certain matrix valued operators which are symmetric with respect to the matrix valued inner product defined by the orthogonality weight. We show that the recurrence coefficients associated with these operators satisfy generalizations of the non-Abelian lattice equations. We provide a Lax pair formulation for these equations, and an example of deformed Hermite-type matrix valued polynomials is discussed in detail.

Non Abelian Toda-type equations and matrix valued orthogonal polynomials

Abstract

In this paper, we study parameter deformations of matrix valued orthogonal polynomials (MVOPs). These deformations are built on the use of certain matrix valued operators which are symmetric with respect to the matrix valued inner product defined by the orthogonality weight. We show that the recurrence coefficients associated with these operators satisfy generalizations of the non-Abelian lattice equations. We provide a Lax pair formulation for these equations, and an example of deformed Hermite-type matrix valued polynomials is discussed in detail.
Paper Structure (12 sections, 15 theorems, 104 equations, 1 figure)

This paper contains 12 sections, 15 theorems, 104 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Fourier algebras of differential and difference operators
  3. Matrix valued deformations of the weight
  4. Symmetry conditions
  5. Toda-type equations
  6. Lax pair
  7. Polynomial Toda equations
  8. Deformed Hermite type polynomials
  9. The deformed weight
  10. Deformed Fourier algebras
  11. Example: the deformed $2\times 2$ weight
  12. Acknowledgements

Key Result

Lemma 3

The coefficient $G_k$ of $M(t)$ is independent of $t$ and $n$.

Figures (1)

  • Figure 1: Scheme for the calculation of $q(M_\Lambda;t)_{\ell}(n;t)$.

Theorems & Definitions (31)

  • Definition 1
  • Remark 2
  • Lemma 3
  • proof
  • Proposition 4
  • proof
  • Remark 5
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 21 more