Discrete Painlevé equations for recurrence coefficients of Laguerre-Hahn orthogonal polynomials

Laguerre-Hahn orthogonal polynomials (LHOP) are related to Stieltjes functions that satisfy a Riccati differential equation with polynomial coefficients. Such families of orthogonal polynomials are very well-known in the literature of special functions and applications. They may be regarded as extensions or generalizations of the so-called classical orthogonal polynomials (Hermite, Laguerre, Jacobi). Indeed, many families of LHOP are obtained through rational spectral transformations of the classical orthogonal polynomials.

In this talk we focus on the problem of deriving properties of orthogonal polynomials and their recurrence coefficients from the knowledge of the polynomials involved in the Riccati equation. It is shown, for some families of LHOP, that the coefficients of the three term recurrence relation satisfy some forms of discrete Painlevé equations, namely,   $dP_I$ and $dP_{IV}$. This is based on joint work with G. Filipuk.