Analytical properties of solutions to nonlinear systems of differential equations associated with some random matrix type models

We obtain new results, as well as those complementing already known ones, concerning the construction of solutions of systems of differential equations corresponding to certain models of random matrix type. These solutions are expressed in terms of solutions of Painlevé II–V equations. We also show that solutions of systems of differential equations associated with random matrix type models having Laguerre and Hermitian kernels satisfy the formal Painlevé test. We obtain new formulas relating solutions of Painlevé III and Painlevé V equations under certain conditions imposed on the parameters entering these equations.