β-Ensembles and higher genera Catalan numbers

We propose formulas for the large N expansion of the generating function of connected correlators of the β-deformed Gaussian and Wishart–Laguerre matrix models. We show that our proposal satisfies the known transformation properties under the exchange of β with 1/β and, using Virasoro constraints, we derive a recursion formula for the coefficients of the expansion. In the undeformed limit β=1, these coefficients are integers and they have the combinatorial interpretation of generalized Catalan numbers. For generic β, we define the higher genus Catalan polynomials Cg,ν(β) whose coefficients are integer numbers.