Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRecurrence coefficients for discrete orthogonal polynomials with hypergeometric weight and discrete Painleve equations
144
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
The classical Painleve equations are so well known that it may come as a surprise to learn that the asymptotic description of its solutions remains incomplete. The problem lies mainly with the description of families of solutions in the complex domain. Where asymptotic descriptions are known, they are stated in the literature as valid for large connected domains, which include movable poles of families of solutions. However, asymptotic analysis necessarily assumes that the solutions are bounded and so these domains must be punctured at locations corresponding to movable poles, leading to asymptotic results that may not be uniformly valid. To overcome these issues, we recently carried out asymptotic analysis in Okamotos geometric space of initial values for the first and second Painleve equations. In this paper, we review this method and indicate how it may be extended to the discrete Painleve equations.
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. We 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.
The Cholesky factorization of the moment matrix is applied to discrete orthogonal polynomials on the homogeneous lattice. In particular, semiclassical discrete orthogonal polynomials, which are built in terms of a discrete Pearson equation, are studied. The Laguerre-Freud structure semi-infinite matrix that models the shifts by $pm 1$ in the independent variable of the set of orthogonal polynomials is introduced. In the semiclassical case it is proven that this Laguerre-Freud matrix is banded. From the well known fact that moments of the semiclassical weights are logarithmic derivatives of generalized hypergeometric functions, it is shown how the contiguous relations for these hypergeometric functions translate as symmetries for the corresponding moment matrix. It is found that the 3D Nijhoff-Capel discrete Toda lattice describes the corresponding contiguous shifts for the squared norms of the orthogonal polynomials. The continuous Toda for these semiclassical discrete orthogonal polynomials is discussed and the compatibility equations are derived. It also shown that the Kadomtesev-Petvishvilii equation is connected to an adequate deformed semiclassical discrete weight, but in this case the deformation do not satisfy a Pearson equation.
In this paper we study a certain recurrence relation, that can be used to generate ladder operators for the Laguerre Unitary ensemble, from the point of view of Sakais geometric theory of Painleve equations. On one hand, this gives us one more detailed example of the appearance of discrete Painleve equations in the theory of orthogonal polynomials. On the other hand, it serves as a good illustration of the effectiveness of a recently proposed procedure on how to reduce such recurrences to some canonical discrete Painleve equations.
We introduce two classes of discrete polynomials and construct discrete equations admitting a Lax representation in terms of these polynomials. Also we give an approach which allows to construct lattice integrable hierarchies in its explicit form and show some examples.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
