Recurrence relations for the generalized Laguerre and Charlier orthogonal polynomials and discrete Painlev\'e equations on the $D_{6}^{(1)}$ Sakai surface

TL;DR

This paper explores how to identify discrete Painlevé equations from recurrence relations, emphasizing the importance of geometric methods and illustrating with examples from orthogonal polynomial theory on Sakai surfaces.

Contribution

It demonstrates the effectiveness of Sakai's geometric approach in identifying discrete Painlevé equations without prior assumptions, using examples from orthogonal polynomials.

Findings

The geometric approach can distinguish non-equivalent translation directions.

Recurrences related to orthogonal polynomials can be regularized on the same Sakai surface.

The identification procedure is effective without prior knowledge of the equation type.

Abstract

This paper concerns the discrete version of the Painlev\'e identification problem, i.e., how to recognize a certain recurrence relation as a discrete Painlev\'e equation. Often some clues can be seen from the setting of the problem, e.g., when the recurrence is connected with some differential Painlev\'e equation, or from the geometry of the configuration of indeterminate points of the equation. The main message of our paper is that, in fact, this only allows us to identify the configuration space of the dynamic system, but not the dynamics themselves. The refined version of the identification problem lies in determining, up to the conjugation, the translation direction of the dynamics, which in turn requires the full power of the geometric theory of Painlev\'e equations. To illustrate this point, in this paper we consider two examples of such recurrences that appear in the theory of orthogonal polynomials. We choose these examples because they get regularized on the same family of Sakai surfaces, but at the same time are not equivalent, since they result in non-equivalent translation directions. In addition, we show the effectiveness of a recently proposed identification procedure for discrete Painlev\'e equations using Sakai's geometric approach for answering such questions. In particular, this approach requires no a priori knowledge of a possible type of the equation.