Recurrence coefficients of generalized Meixner polynomials and Painlevé equations

Abstract

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 dPIV 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 PV.

The generalized Krawtchouk polynomials and the fifth Painlevé equation

“…This is similar to the case of generalized Meixner polynomials (see [2]). By taking y n ðcÞ in a form as shown in (14), we get the fifth Painlevé equation P V (15) with parameters (16).…”

“…; N}). For more information about discrete (classical) orthogonal polynomials, we refer to [19,26] (see also [2,9,11] for the definitions of classical weights and their semi-classical extensions and further references).…”

“…2 or deg t -1 (see [18,23]). The recurrence coefficients in the three-term recurrence relation (2) for classical orthogonal polynomials (e.g. Hermite, Laguerre, Jacobi, Charlier, Meixner, Krawtchouk and others) can be found explicitly in contrast to non-classical weights.…”

“…One can, similarly to [2] (Section 3), check that the functions y n21 , y n and y nþ1 in (41) are connected by using the Bäcklund transformation of the fifth Painlevé equation ([8], §32.7 (v)). In particular, one can express all of them using only y and y 0 (the solution of P V with parameters (16)).…”

Ladder operators and differential equations for multiple orthogonal polynomials

“…On the other hand, ladder operators has been successfully applied to establish the connections between Painlevé equations and recurrence coefficients of certain orthogonal polynomials; cf. [6,9,13,14,19,21,22,23] for recent applications.…”