Related papers: Remarks on $\tau$-functions for the difference Pai…
Let v be a real polynomial of even degree, and let \rho be the equilibrium probability measure for v with support S; so that v(x)\geq 2\int \log |x-y| \rho (dy)+C_v for some constant C_v with support S. Then S is the union of finitely many…
In our previous works, a relationship between Hermite's two approximation problems and Schlesinger transformations of linear differential equations has been clarified. In this paper, we study tau-functions associated with holonomic…
In this letter we establish a connection of Picard-type elliptic solutions of Painleve VI equation with the special solutions of the non-stationary Lame equation. The latter appeared in the study of the ground state properties of Baxter's…
The Painleve test is very useful to construct not only the Laurent-series solutions but also the elliptic and trigonometric ones. Such single-valued functions are solutions of some polynomial first order differential equations. To find the…
We study the $n$-point differentials corresponding to Kadomtsev-Petviashvili tau functions of hypergeometric type (also known as Orlov-Scherbin partition functions), with an emphasis on their $\hbar^2$-deformations and expansions. Under the…
We review recent results on the finite-gap properties of difference operators with elliptic coefficients and give explicit characterization of spectral curves for difference analogues of the higher Lam\'e operators. This curve parametrizes…
In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlev\'e equations, with a particular emphasis on the discrete Painlev\'e equations. The theory is controlled by the…
The Painleve test is very useful to construct not only the Laurent series solutions of systems of nonlinear ordinary differential equations but also the elliptic and trigonometric ones. The standard methods for constructing the elliptic…
We continue the study of the B-Toda hierarchy (the Toda lattice with the constraint of type B) which can be regarded as a discretization of the BKP hierarchy. We introduce the tau-function of the B-Toda hierarchy and obtain the bilinear…
We introduce hierarchies of difference equations (referred to as $nT$-systems) associated to the action of a (centrally extended, completed) infinite matrix group $GL_{\infty}^{(n)}$ on $n$-component fermionic Fock space. The solutions are…
Matrix elements in different representations are connected by quadratic relations. If matrix elements are those of a $\textit{group element}$, i.e. satisfying the property $\Delta(X) = X\otimes X$, then their generating functions obey…
We discuss some new aspects of the theory of the Jimbo-Miwa-Ueno tau function which have come to light within the recent developments in the global asymptotic analysis of the tau functions related to the Painlev\'e equations. Specifically,…
In this paper we develop the calculus of pseudo-differential operators on the lattice $\mathbb{Z}^n$, which we can call pseudo-difference operators. An interesting feature of this calculus is that the phase space is compact so the symbol…
We consider certain examples of applications of the general methods, based on geometry and integrability of matrix models, described in hep-th/0601212. In particular, the nonlinear differential equations, satisfied by quasiclassical…
A new Lax pair for the sixth Painlev\'e equation $P_{VI}$ is constructed in the framework of the loop algebra $\mathfrak{so}(8)[z,z^{-1}]$. The whole affine Weyl group symmetry of $P_{VI}$ is interpreted as gauge transformations of the…
This paper presents the basic ideas and properties of elliptic functions and elliptic integrals as an expository essay. It explores some of their numerous consequences and includes applications to some problems such as the simple pendulum,…
In this paper, we investigate the Euler-type integral representations for the generalized hypergeometric matrix function and develop some transformations in terms of hypergeometric matrix functions. Furthermore, unit and half arguments have…
We extend two of the methods previously introduced to find discrete symmetries of differential equations to the case of difference and differential-difference equations. As an example of the application of the methods, we construct the…
The paper is an essentially extended version of the work math.CA/0601371, supplemented with an application. We present new results in the theory of classical $\theta$-functions of Jacobi and $\sigma$-functions of Weierstrass: ordinary…
In these lecture notes I give an elementary introduction to elliptic hypergeometric functions. I focus on motivating the main ideas and constructions, rather than giving a comprehensive survey. The lectures include a brief explanation of…