Meixner Matrix Ensembles

Abstract

Abstract. 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+ cI n where X and Y are i.i.d. on 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.

Equivalences of Linear Functional Systems

“…which is studied in probability theory [2]. Let D := Q(β)(x 1 , x 2 ) ∂ 1 , ∂ 2 be the noncommutative ring of PD operators in ∂ 1 := ∂ ∂x1 and ∂ 2 := ∂ ∂x2 with coefficients in the field Q(β, x 1 , x 2 ) of rational functions in x 1 , x 2 , and β.…”

Josef Meixner: His life and his orthogonal polynomials

“…In principle, another kind of multivariable Meixner polynomials, associated with root system BC n , can be obtained as a limit case of the BC n Hahn polynomials considered by van Diejen & Stokman [46,Section 5]. Possibly the papers by Chikuse [43] and Bryc & Letac [39], written from a probabilistic point of view, and the paper by Shibukawa [75] are related.…”