On differential systems related to generalized Meixner and deformed Laguerre orthogonal polynomials

In this paper we present a connection between systems of differential equations for the recurrence coefficients of polynomials orthogonal with respect to the generalized Meixner and the deformed Laguerre weights. It is well-known that the recurrence coefficients of both generalized Meixner and deformed Laguerre orthogonal polynomials can be expressed in terms of solutions of the fifth Painlevé equation but no explicit relation between systems of differential equations for the recurrence coefficients was known. We also present certain limits in which the recurrence coefficients can be expressed in terms of solutions of the Painlevé XXXIV equation, which in the deformed Laguerre case extends previous studies and in the generalized Meixner case is a new result