Over the last decade it has become clear that discrete Painlevé equations appear in a wide range of important mathematical and physical problems. Thus, the question of recognizing a given non-autonomous recurrence as a discrete Painlevé equation and determining its type according to Sakai's classification scheme, understanding whether it is equivalent to some known (model) example, and especially finding an explicit change of coordinates transforming it to such an example, becomes one of the central ones. Fortunately, Sakai's geometric theory provides an almost algorithmic procedure for answering this question. In this paper we illustrate this procedure by studying an example coming from the theory of discrete orthogonal polynomials. There are many connections between orthogonal polynomials and Painlevé equations, both differential and discrete. In particular, often the coefficients of three-term recurrence relations for discrete orthogonal polynomials can be expressed in terms of solutions of discrete Painlevé equations. In this work we study discrete orthogonal polynomials with general hypergeometric weight and show that their recurrence coefficients satisfy, after some change of variables, the standard discrete Painlevé-V equation. We also provide an explicit change of variables transforming this equation to the standard form.
We study singularity confinement phenomena in examples of delay-differential Painlevé equations, which involve shifts and derivatives with respect to a single independent variable. We propose a geometric interpretation of our results in terms of mappings between jet spaces, defining certain singularities analogous to those of interest in the singularity analysis of discrete systems, and what it means for them to be confined. For three previously studied examples of delay-differential Painlevé equations, we describe all such singularities and show they are confined in the sense of our geometric description.
Since the classification of discrete Painlevé equations in terms of rational surfaces, there has been much interest in the range of integrable equations arising from each of the 22 surface types in Sakai's list. For all but the most degenerate type in the list, the surfaces come in families which admit affine Weyl groups of symmetries. Translation elements of this symmetry group define discrete Painlevé equations with the same number of parameters as their family of surfaces. While non-translation elements of the symmetry group have been observed to correspond to discrete systems of Painlevé-type through projective reduction, these have fewer than the maximal number of free parameters corresponding to their surface type. We show that difference equations with the full number of free parameters can be constructed from non-translation elements of infinite order in the symmetry group, constructing several examples and demonstrating their integrability. This is prompted by the study of a previously proposed discrete Painlevé equation related to a special class of discrete analogues of surfaces of constant negative Gaussian curvature, which we generalise to a full-parameter integrable difference equation, given by the Cremona action of a non-translation element of the extended affine Weyl group W (D (1) 4 ) on a family of generic D (1) 4 -surfaces.2
In this paper we present a connection between systems of differential equations for the recurrence coefficients of polynomials orthogonal with respect to the generalized Meixner and the deformed Laguerre weights. It is well-known that the recurrence coefficients of both generalized Meixner and deformed Laguerre orthogonal polynomials can be expressed in terms of solutions of the fifth Painlevé equation but no explicit relation between systems of differential equations for the recurrence coefficients was known. We also present certain limits in which the recurrence coefficients can be expressed in terms of solutions of the Painlevé XXXIV equation, which in the deformed Laguerre case extends previous studies and in the generalized Meixner case is a new result.
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