Polynomial Transformations of Jacobi Matrices, Matrix valued Orthogonal polynomials and Random Walks

Abstract

In this paper, we study matrix valued orthogonal polynomials whose block Jacobi matrix is $\Theta (J)$, viewed in blocks of size $m=\mathrm{deg}\Theta$, where $J$ is a tridiagonal Jacobi matrix and $\Theta$ is a scalar polynomial. We identify their matrix measure of orthogonality intrinsically as the push-forward, under $\Theta$, of a matrix measure constructed from the original scalar orthogonal polynomials. We formulate reducibility in the push-forward variable and prove that the $2\times 2$ Hermite family associated with $\Theta (x)=x^2+{\alpha }_{1}x$ is irreducible for ${\alpha }_1\ne 0$ and reducible for ${\alpha }_1=0$. This problem is motivated by random walks with transition matrix $\Theta (J)$. For these models, we derive scalar and block Karlin--McGregor representations for the powers of $\Theta (J)$. We apply the construction to the mixed one- and two-ball Ehrenfest chain, obtaining explicit spectral measures, spectral multiplicities, and transition formulas, and we conclude with the corresponding $k$-ball and multi-ball extensions.