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

Abstract

A general family of matrix valued Hermite type orthogonal polynomials is introduced as the matrix orthogonal polynomials with respect to a weight. The matrix polynomials are eigenfunctions of a matrix differential equation. For the weight we derive Pearson equations, which allow us to derive many explicit properties of these matrix polynomials. In particular, the matrix polynomials are eigenfunctions to another matrix differential equation. We also obtain for these polynomials shift operators, a Rodrigues formula, explicit expressions for the squared norm, explicit three term recurrence relations, etc. The matrix entries of the matrix polynomials can be expressed in terms of scalar Hermite and dual Hahn polynomials. We also derive a connection formula for the matrix Hermite polynomials. Next we show that operational Burchnall formulas extend to matrix polynomials. We make this explicit for the matrix Hermite polynomials and for previously introduced matrix Gegenbauer type orthogonal polynomials. The Burchnall approach gives two descriptions of the matrix valued orthogonal polynomials for the Toda modification of the matrix Hermite weight. In particular, we obtain an explicit non-trivial solution to the non-abelian Toda lattice equations.