A recursion formula for mixed trace moments of isotropic Wishart matrices and the Gaussian unitary ensemble

Recursion formulas for mixed moments of three fundamental random matrix ensembles are derived. The reason such recursive formulas are possible is closely related to properties of polygon gluings studied by Akhmedov and Shakirov. The proofs of the formulas are however written in such a way that they do not rely on such polygon-models and can be understood without a background in combinatorics.