Expansions of solutions to the sixth Painlevé equation in the cases a = 0 and b = 0

Solving the Sixth Painlevé Equation: Towards the Classification of all the Critical Behaviors and the Connection Formulae

“…The coincidence of the result of the present paper with that of the elliptic representation, together with the convergence of the expansions of δ * E (x), proves the convergence of (18) and (26). 4 Proof of Proposition 1. The critical behavior at x = 0 when ℜσ = 1.…”

“…Among them, the works of S.Shimomura (the results are summarized in [20]) and A.D.Bruno, I.V. Goryuchkina ( [2] [3] [4] [5] [6]) are local approaches, which do not determine the connection formulae, but essentially determine all the critical behaviors or asymptotic expansions. In [14] [15] [16] [17], in the framework of the method of 2 H. Umemura proved the of irreducibility of the Painlevé equations [30] [31] [32].…”

“…All the algebraic solutions were classified in [9] when β = γ = 0, δ = 1 2 , and then in [24] for the general PVI. 3 A more restrictive definition of "solving" should include the distribution of the poles (movable singularities) of the transcendents. This problem for PVI is still open (in [14], the behavior on the universal covering of a critical point is analyzed and it is shown that if the poles exist, they are distributed in spirals converging to the critical point).…”

The logarithmic asymptotics of the sixth Painlevé equation

“…y(x) ∼ a x 1−σ , if ℜσ > 0; (5) y(x) ∼ x iA sin iσ ln x + φ + θ 2 0 − θ 2 x + σ 2 2σ 2 , if ℜσ = 0, σ = 0.…”

“…8.1 Action on the Transcendent. Formulae (11), (12), (13), (14) The symmetry σ 3 , acting on the transcendent (6), gives the behavior:…”