Meixner Matrix Ensembles

We construct a family of matrix ensembles that fits Anshelevich’s regression postulates for “Meixner laws on matrices,” namely the distribution with an invariance property of X when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{E}(\mathbf {X}^{2}|\mathbf {X}+\mathbf {Y})=a(\mathbf {X}+\mathbf {Y})^{2}+b(\mathbf {X}+\mathbf {Y})+c\mathbf {I}_{n}$\end{document} where X and Y are i.i.d. symmetric matrices of order n. We show that the Laplace transform of a general n×n Meixner matrix ensemble satisfies a system of partial differential equations which is explicitly solvable for n=2. We rely on these solutions to identify the six types of 2×2 Meixner matrix ensembles.