On functions of Jacobi-Weierstrass (I) and equation of Painleve

TL;DR

This paper extends the theory of Jacobi and Weierstrass functions by deriving new differential equations and series expansions, and applies these results to explicitly represent solutions of the sixth Painlevé equation.

Contribution

It introduces new results on classical theta and sigma functions, extending their canonical forms and applying them to Painlevé VI solutions.

Findings

Derived new differential equations for theta and sigma functions

Provided series expansions for these functions

Explicitly represented Painlevé VI solutions in terms of tau-functions

Abstract

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 differential equations and series expansions. We also give the extension of canonical $\theta$-functions and consider an application to the sixth Painlev\'e equation (P6). Picard--Hitchin's general solution of P6 is represented explicitly in a form of logarithmic derivative of a corresponding $\tau$-function (Painlev\'e's form).