Mixed correlation functions of the two-matrix model

We compute the correlation functions mixing the powers of two non-commuting random matrices within the same trace. The angular part of the integration was partially known in the literature (Morozov A 1992 Mod. Phys. Lett. A 7 3503, Shatashvili S L 1993 Commun. Math. Phys. 154 421): we pursue the calculation and carry out the eigenvalue integration reducing the problem to the construction of the associated bi-orthogonal polynomials. The generating function of these correlations then becomes a determinant involving the recursion coefficients of the bi-orthogonal polynomials.