# Moments of discrete orthogonal polynomial ensembles

## TL;DR

This paper derives factorial moment identities for discrete orthogonal polynomial ensembles like Charlier, Meixner, and Krawtchouk, revealing polynomial properties and recurrence relations that connect to equilibrium measures and Schur measures.

## Contribution

It provides new hypergeometric representations and recurrence relations for factorial moments of these ensembles, extending previous work and linking to Schur measure identities.

## Key findings

- Factorial moments expressed via hypergeometric functions.
- Recurrence relations and differential equations for moments.
- Connection between moments and equilibrium measure properties.

## 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.