Picard and Chazy solutions to the Painlevé VI equation

Abstract

Abstract. I study the solutions of a particular family of Painlevé VI equations with the parameters β = γ = 0, δ = 1 2 and 2α = (2µ − 1) 2 , for 2µ ∈ Z Z. I show that the case of half-integer µ is integrable and that the solutions are of two types: the so-called Picard solutions and the so-called Chazy solutions. I give explicit formulae for them and completely determine their asymptotic behaviour near the singular points 0, 1, ∞ and their nonlinear monodromy. I study the structure of analytic continuation of the solutions to the PVIµ equation for any µ such that 2µ ∈ Z Z. As an application, I classify all the algebraic solutions. For µ half-integer, I show that they are in one to one correspondence with regular polygons or star-polygons in the plane. For µ integer, I show that all algebraic solutions belong to a one-parameter family of rational solutions.SISSA preprint no. 89/98/FM,

Holonomy of the Ising model form factors

“…These identities, now, become remarkable identities on some infinite Gaussian sums, or on For N = 0 this equation has been solved in terms of theta functions [40,41,42], has dihedral symmetry and has a countable number of algebraic solutions. ♯ To be considered when comparing with [40].…”

The eight-vertex model and Painlevé VI

“…Note that such parameterization of Picard's solutions of P V I with the above choice (48) of the parameters c 1 and c 2 has already appeared in [30,31].…”

Fifth Painleve Transcendents Which are Analytic at the Origin

“…As far as we know, this was the first example of calculation of the linear monodromy of Painlevé functions. We also remark that the main result of [4] was recently rediscovered by [14].…”