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Recursion formulas for mixed moments of three fundamental random matrixensembles are derived. The reason such recursive formulas are possible isclosely related to properties of polygon gluings studied by Akhmedov andShakirov. The proofs of the formulas are however written in such a way thatthey do not rely on such polygon-models and can be understood without abackground in combinatorics.

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