Abstract
We obtain factorial moment identities for the Charlier, Meixner and Krawtchouk orthogonal polynomial ensembles. Building on earlier results by Ledoux [Elect. J. Probab. 10, (2005)], we find hypergeometric representations for the factorial moments when the reference measure is Poisson (Charlier ensemble) and geometric (a particular case of the Meixner ensemble). In these cases, if the number of particles is suitably randomised, the factorial moments have a polynomial property, and satisfy three-term recurrence relations and differential equations. In particular, the normalised factorial moments of the randomised ensembles are precisely related to the moments of the corresponding equilibrium measures. We also briefly outline how these results can be interpreted as Cauchy-type identities for certain Schur measures.
Highlights
Introduction and definitionsAn orthogonal polynomial ensemble is a probability measure on RN given by dQ(x) = ∆(x)2 N dμ(xj ), j=1 (1.1)where x = (x1, . . . , xN ), μ is a probability measure on the real line having all moments,∆(x) = det 1≤i,j≤N xji −1(xj − xi), i
There are a number of models from statistical physics, probability theory and combinatorics, which are described in terms of orthogonal polynomial ensembles
The measure (1.1) can be conveniently analysed by using the so-called orthogonal polynomial method pioneered by Mehta [42] in the study of random matrices
Summary
Theorems 1, 2 and 3 are the starting points of our analysis on the factorial moments of the classical discrete OP ensembles as functions of k. We found similar hypergeometric series for the factorial moments of the classical discrete OP ensembles when the reference measure is Poisson μθ(x) = θxe−θ/x!, θ > 0 (Charlier ensemble) and geometric μ1q(x) = qx(1 − q) (Meixner ensemble with parameter γ = 1). The factorial moment of the Charlier ensemble can be written as. The factorial moment of the Meixner ensemble with γ = 1 can be written as. Mq1(k, extended to the entire complex plane to a polynomial in k of degree 2(N − 1), and when k ∈ N, it is a polynomial in (N − 1) of degree k. Q 1−q q 1−q q 1−q k (2)k k + 1 k k2 + kq + 2q (3)k (k + 1)(k + 2)q k k4 + 4k3q − 2k3 + 2k2q2 + 8k2q + k2 + 10kq2 + 12q2
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