  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.
