Related papers: Remarks on $\tau$-functions for the difference Pai…

We study explicit formula (suggested by Gamayun, Iorgov, Lisovyy) for Painlev\'e III($D_8$) $\tau$ function in terms of Virasoro conformal blocks with central charge $1$. The Painlev\'e equation has two types of bilinear forms, we call them…

We introduce the concept of $\omega$-lattice, constructed from $\tau$ functions of Painlev\'e systems, on which quad-equations of ABS type appear. In particular, we consider the $A_5^{(1)}$- and $A_6^{(1)}$-surface $q$-Painlev\'e systems…

In recent years, the Fourier series (Zak transform) structure of the Painlev\'e I tau function has emerged in multiple contexts. Its main building block admits several conjectural interpretations, such as the partition function of an…

We derive determinant representations and nonlinear differential equations for the scaled 2-point functions of the 2D Ising model on the cylinder. These equations generalize well-known results for the infinite lattice (Painlev\'e III…

In this paper we construct explicit solutions and calculate the corresponding $\tau$-function to the system of Schlesinger equations describing isomonodromy deformations of $2\times 2$ matrix linear ordinary differential equation whose…

In this paper, we construct two lattices from the $\tau$ functions of $A_4^{(1)}$-surface $q$-Painlev\'e equations, on which quad-equations of ABS type appear. Moreover, using the reduced hypercube structure, we obtain the Lax pairs of the…

We extend a recent result of [13] for the KdV hierarchy to the Toda lattice hierarchy. Namely, for an arbitrary solution to the Toda lattice hierarchy, we define a pair of wave functions, and use them to give explicit formulae for the…

We introduce and study symmetric polynomials, which as very special cases include polynomials related to the supersymmetric eight-vertex model, and other elliptic lattice models with $\Delta=\pm 1/2$. There is also a close relation to…

Painleve's transcendental differential equation P_{VI} may be expressed as the consistency condition for a pair of linear differential equations with 2 by 2 matrix coefficients with rational entries. By a construction due to Tracy and…

Based on the matrix-resolvent approach, for an arbitrary solution to the discrete KdV hierarchy, we define the tau-function of the solution, and compare it with another tau-function of the solution defined via reduction of the Toda lattice…

In this paper we obtain large $z$ asymptotic expansions in the complex plane for the tau function corresponding to special function solutions of the Painlev\'e II differential equation. Using the fact that these tau functions can be written…

We present a general theory for studying the difference analogues of special functions of hypergeometric type on the linear-type lattices, i.e., the solutions of the second order linear difference equation of hypergeometric type on a…

In a previous paper, a point of order 8 on an elliptic curve was calculated. Exploiting the well-known correspondence of the points on an elliptic curve with the points of a respective period parallelogram, we proceed to calculating all…

Building upon the recent works of Bertola; Fasondini, Olver and Xu, we define a class of orthogonal polynomials on elliptic curves and establish a corresponding Riemann-Hilbert framework. We then focus on the special case, defined by a…

We establish a bilinear framework for elliptic soliton solutions which are composed by the Lam\'e-type plane wave factors. $\tau$ functions in Hirota's form are derived and vertex operators that generate such $\tau$ functions are presented.…

We introduce the tau-function of a rational d-connection and its isomonodromy transformations. We show that in a continuous limit our tau-function agrees with the Jimbo-Miwa-Ueno tau-function, compute the tau-function for the isomonodromy…

We first review the properties of the conventional $\tau$-functions of the KP and Toda-lattice hierarchies. A straightforward generalization is then discussed. It corresponds to passing from differential to finite-difference equations; it…

Using the matrix-resolvent method and a formula of the second-named author on the $n$-point function for a KP tau-function, we show that the tau-function of an arbitrary solution to the Toda lattice hierarchy is a KP tau-function. We then…

By analogy to the continuous Painlev\'e II equation, we present particular solutions of the discrete Painlev\'e II (d-P$\rm_{II}$) equation. These solutions are of rational and special function (Airy) type. Our analysis is based on the…

We present some observations on the tau-function for the fourth Painlev\'e equation. By considering a Hirota bilinear equation of order four for this tau-function, we describe the general form of the Taylor expansion around an arbitrary…