Recurrence coefficients of semi-classical orthogonal polynomials are often related to the solutions of special nonlinear second-order differential equations known as the Painlevé equations. Each Painlevé equation can be written in a standard form as a non-autonomous Hamiltonian system, so it is natural to ask whether differential systems satisfied by the recurrence coefficients also possess Hamiltonian structures. We consider recurrence coefficients for a modified Laguerre weight which satisfy a differential system known to be related to the modified third Painlevé equation and identify a Hamiltonian structure for it by constructing its space of initial conditions. We also discuss a transformation from this system to the modified third Painlevé equation which simultaneously identifies a discrete system for the recurrence coefficients with a discrete Painlevé equation
