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Exact recursion formulas for mixed moments of four fundamental random matrixensembles are derived. The reason such recursive formulas are possible isclosely related to properties of polygon gluings studied by Harer and Zagier aswell as Akhmedov and Shakirov. The proofs of the formulas are direct andwritten in such a way that they do not rely on understanding of polygongluings.

A recursion formula for mixed trace moments of isotropic Wishart matrices and the Gaussian unitary/orthogonal ensembles