Related papers: Discrete Painlev\'e Equations
We present particular solutions of the discrete Painlev\'e III (d-P$\rm_{III}$) equation of rational and special function (Bessel) type. These solutions allow us to establish a close parallel between this discrete equation and its…
Exact solutions of the dispersive and modified equations are expressed in terms of special polynomials associated with rational solutions of the fourth Painleve equation, which arises as generalized scaling reductions of these equations.…
We study the degenerate Garnier system which generalizes the fifth Painlev\'{e} equation. We present two classes of particular solutions, classical transcendental and algebraic ones. Their coalescence structure is also investigated.
We revisit the enumeration problems of random discrete surfaces (RDS) based on solutions of the discrete equations derived from the matrix models. For RDS made of squares, the recursive coefficients of orthogonal polynomials associated with…
The purpose of this article is to motivate the study of invariant, and especially conformally invariant, differential pairings. Since a general theory is lacking, this work merely presents some interesting examples of these pairings,…
We study the set of algebraic numbers of bounded height and bounded degree where an analytic transcendental function takes algebraic values.
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…
We represent and analyze the general solution of the sixth Painleve transcendent in the Picard-Hitchin-Okamoto class in the Painleve form as the logarithmic derivative of the ratio of certain $\tau$-functions. These functions are…
The main purpose of this article is concerned with the existence and the precise forms of the transcendental solutions of several refined versions of Fermat-type functional equations with polynomial coefficients in several complex variables…
The incomplete version of the Macdonald function has various appellations in literature and earns a well-deserved reputation of being a computational challenge. This paper ties together the previously disjoint literature and presents the…
Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics…
Probabilistic solvers for ordinary differential equations (ODEs) have emerged as an efficient framework for uncertainty quantification and inference on dynamical systems. In this work, we explain the mathematical assumptions and detailed…
The main goal of this paper is to provide a brief survey of recent results which connect together results from different areas of research. It is well known that numerical integration of functions with mixed smoothness is closely related to…
Over the past two decades, the field of high-dimensional statistics has experienced substantial progress, driven largely by technological advances that have dramatically reduced the cost and effort for data collection and storage across a…
We count algebraic points of bounded height and degree on the graphs of certain functions analytic on the unit disk, obtaining a bound which is polynomial in the degree and in the logarithm of the multiplicative height. We combine this work…
We introduce and discuss a new class of (multivalued analytic) transcendental functions which still share with algebraic functions the property that the number of their isolated zeros can be explicitly counted. On the other hand, this class…
We investigate the discrete Painleve II equation over finite fields. We treat it over local fields and observe that it has a property that is similar to the good reduction over finite fields. We can use this property, which seems to be an…
This paper introduces a new functional expansion framework that extends classical ideas beyond the Taylor series. Unlike traditional Taylor expansions based on local polynomial approximations, the proposed approach arises from exact…
In this paper, we consider the discrete power function associated with the sixth Painlev\'e equation. This function is a special solution of the so-called cross-ratio equation with a similarity constraint. We show in this paper that this…
In this paper we present an abstraction-refinement approach to Satisfiability Modulo the theory of transcendental functions, such as exponentiation and trigonometric functions. The transcendental functions are represented as uninterpreted…