Recurrence coefficients of generalized Meixner polynomials and Painlevé equations

We consider a semi-classical version of the Meixner weight depending on two parameters and the associated set of orthogonal polynomials. These polynomials satisfy a three-term recurrence relation. We show that the coefficients appearing in this relation satisfy a discrete Painlevé equation, which is a limiting case of an asymmetric dP IV equation. Moreover, when viewed as functions of one of the parameters, they satisfy one of Chazy's second-degree Painlevé equations, which can be reduced to the fifth Painlevé equation P V .