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**ABSTRACT:**We represent and analyze the general solution of the sixth Painlevé transcendent $ \mathcal{P}_6 $ \mathcal{P}_6 in the Picard-Hitchin-Okamoto class in the Painlevé form as the logarithmic derivative of the ratio of τ-functions. We express these functions explicitly in terms of the elliptic Legendre integrals and Jacobi theta functions, for which we write the general differentiation rules. We also establish a relation between the $ \mathcal{P}_6 $ \mathcal{P}_6 equation and the uniformization of algebraic curves and present examples.Theoretical and Mathematical Physics 12/2010; 161(3):1616-1633. · 0.70 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We study the analytic properties and the critical behavior of the elliptic representation of solutions of the Painlev\'e 6 equation. We solve the connection problem for elliptic representation in the generic case and in a non-generic case equivalent to WDVV equations of associativity.Communications on Pure and Applied Mathematics 07/2002; 55(10):1280 - 1363. · 3.34 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We will describe a method for constructing explicit algebraic solutions to the sixth Painleve equation, generalising that of Dubrovin-Mazzocco. There are basically two steps: First we explain how to construct finite braid group orbits of triples of elements of SL_2(C) out of triples of generators of three-dimensional complex reflection groups. (This involves the Fourier-Laplace transform for certain irregular connections.) Then we adapt a result of Jimbo to produce the Painleve VI solutions. (In particular this solves a Riemann-Hilbert problem explicitly.) Each step will be illustrated using the complex reflection group associated to Klein's simple group of order 168. This leads to a new algebraic solution with seven branches. We will also prove that, unlike the algebraic solutions of Dubrovin-Mazzocco and Hitchin, this solution is not equivalent to any solution coming from a finite subgroup of SL_2(C). The results of this paper also yield a simple proof of a recent theorem of Inaba-Iwasaki-Saito on the action of Okamoto's affine D4 symmetry group as well as the correct connection formulae for generic Painleve VI equations.Proceedings of the London Mathematical Society 09/2003; · 1.12 Impact Factor

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