Related papers: A tau-function solution to the sixth Painleve tran…

In families of Painlev\'e VI differential equations having common algebraic solutions we classify all the members which come from geometry, i.e. the corresponding linear differential equations which are Picard-Fuchs associated to families…

This expository article written for the Notices of the American Mathematical Society provides an overview of transcendental functions arising as solutions of the discrete Painlev\'e equations, for which the developments of the last two…

This paper first discusses irreducibility of a Painlev\'e equation $P$. We explain how the Painlev\'e property is helpful for the computation of special classical and algebraic solutions. As in a paper of Morales-Ruiz we associate an…

Several results on Heun's equation are generalized to a certain class of Fuchsian differential equations. Namely, we obtain integral representations of solutions and develop Hermite-Krichever Ansatz on them. In particular, we investigate…

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…

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 classify all values of the parameters $\alpha$, $\beta$, $\gamma$ and $\delta$ of the Painlev\'e VI equation such that there are rational solutions. We give a formula for them up to the birational canonical transformations…

For an arbitrary solution to the KdV hierarchy, the generating series of logarithmic derivatives of the tau-function of the solution can be expressed by the basic matrix resolvent via algebraic manipulations. Based on this we develop in…

In this paper we find fractional Riemann-Liouville derivatives for the Takagi-Landsberg functions. Moreover, we introduce their generalizations called weighted Takagi-Landsberg functions which have arbitrary bounded coefficients in the…

We evaluate the total integral from negative infinity to positive infinity of all global solutions to the Painleve II equation on the real line. The method is based on the interplay between one of the equations of the associated Lax pair…

We introduce a notion of the divisor type for rational functions and show that it can be effectively used for the classification of the deformations of dessins d'enfants related with the construction of the algebraic solutions of the sixth…

The problem of Painleve classification of ordinary differential equations lasting since the end of XIX century saw significant advances for the limited equation order, however not that much for the equations of higher orders. In this work…

We present a class of solutions to the discrete Painlev\'e-II equation for particular values of its parameters. It is shown that these solutions can be expressed in terms of Casorati determinants whose entries are discrete Airy functions.…

This paper continues our previous work done in math.AG/0008207 and is an attempt to establish a conceptual framework which generalizes the work of Manin on the relation between non-linear second order ODEs of type Painleve VI and integrable…

The relation between the Painleve equations and the algebraic equations with the catastrophe theory point of view are considered. The asymptotic solutions with respect to the small parameter of the Painleve equations different types are…

We derive series representations for the tau functions of the $q$-Painlev\'e V, $\mathrm{III_1}$, $\mathrm{III_2}$, and $\mathrm{III_3}$ equations, as degenerations of the tau functions of the $q$-Painlev\'e VI equation in [Jimbo M., Nagoya…

In this paper, usual Sturm-Liouville problems are extended for symmetric functions so that the corresponding solutions preserve the orthogonality property. Two basic examples, which are special cases of a generalized Sturm-Liouville…

The first part of the paper is devoted to two descriptions of all polynomial tau-functions of the KP hierarchy: by a generalized Jacobi-Trudy formula, and a generalized Giambelli formula. We use the latter formula in the second part to…

We will write down the higher genus algebraic curves supporting icosahedral solutions of the sixth Painleve equation, including the largest (genus seven) curve.

We study a special anisotropic XYZ-model on a periodic chain of an odd length and conjecture exact expressions for certain components of the ground state eigenvectors. The results are written in terms of tau-functions associated with…