In this paper we present a general scheme for how to relate differential equations for the recurrence coefficients of semi-classical orthogonal polynomials to the Painleve equations using the geometric framework of Okamotos space of initial values. W
e demonstrate this procedure in two examples. For semi-classical Laguerre polynomials appearing in [HC17], we show how the recurrence coefficients are connected to the fourth Painleve equation. For discrete orthogonal polynomials associated with the hypergeometric weight appearing in [FVA18] we discuss the relation of the recurrence coefficients to the sixth Painleve equation. In addition to demonstrating the general scheme, these results supplement previous studies [DFS20, HFC20], and we also discuss a number of related topics in the context of the geometric approach, such as Hamiltonian forms of the differential equations for the recurrence coefficients, Riccati solutions for special parameter values, and associated discrete Painleve equations.
It is well-known that differential Painleve equations can be written in a Hamiltonian form. However, a coordinate form of such representation is far from unique -- there are many very different Hamiltonians that result in the same differential Painle
ve equation. In this paper we describe a systematic procedure of finding changes of coordinates transforming different Hamiltonian systems into some canonical form. Our approach is based on Sakais geometric theory of Painleve equations. We explain our approach using the fourth differential ${text{P}_{mathrm{IV}}}$ equation as an example, but it can be easily adapted to other Painleve equations as well.
Over the last decade it has become clear that discrete Painleve 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 Painleve equation and
determining its type according to Sakais 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, Sakais 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 Painleve 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 Painleve 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 Painleve-V equation. We also provide an explicit change of variables transforming this equation to the standard form.
