Elements of proof for conjectures of Witte and Forrester about the combinatorial structure of Gaussian β Ensembles

The purpose of the article is to provide partial proofs for two conjectures given by Witte and Forrester in "Moments of the Gaussian beta Ensembles and the large N expansion of the densities" ([1]) with the use of the topological recursion adapted for general beta Gaussian case. In particular, the paper uses a version at coinciding points that provides a simple proof for some of the coefficients involved in the conjecture. Additionally, we propose a generalized version of the conjectures for all correlation functions evaluated at coinciding points.