Transformations RS 4 2 (3) of the Ranks ≤ 4¶and Algebraic Solutions of the Sixth Painlevé Equation

Abstract

Compositions of rational transformations of independent variables of linear matrix ordinary differential equations (ODEs) with the Schlesinger transformations (RS-transformations) are used to construct algebraic solutions of the sixth Painlevé equation. RS-Transformations of the ranks 3 and 4 of 2 × 2 matrix Fuchsian ODEs with 3 singular points into analogous ODE with 4 singular points are classified.

R. Fuchs’ problem of the Painlevé equations from the first to the fifth

“…In the case β = 0, δ = 0 (or α = 0, γ = 0), the third Painlevé equation is a quadrature, and we exclude this case from the Painlevé family. D (1) j (j = 6, 7, 8) mean the affine Dynkin diagrams corresponding to Okamoto's initial value spaces. In this paper we omit the upper index (1) for simplicity and denote D 6 , D 7 , D 8 .…”

On the reductions and classical solutions of the Schlesinger equations

“…all isomonodromic deformations of the Fuchsian system are given by solutions to the Schlesinger equations (see, e.g., [36]) 1 . The solutions to the Schlesinger equations can be parameterized by the monodromy data of the Fuchsian system (1.2) (see precise definition below in Section 2).…”

“…This is a generalization of the celebrated Painlevé property of absence of movable critical singularities (see details in [14,15]). In certain cases the technique based on the theory of Riemann-Hilbert problem gives a possibility 1 Bolibruch constructed non-Schlesinger isomonodromic deformations in [4]. These can occur when the matrices A i are resonant, i.e.…”

“…This problem remains open even for the case of Painlevé-VI although there are interesting results based on the theory of symmetries of Painlevé equations [34,33,1] and on the geometric approach to studying the space of monodromy data [7,13,27,28].…”

Some Examples of RS23(3)-Transformations of Ranks 5 and 6 as the Higher Order Transformations for the Hypergeometric Function

“…In the work [4] it was shown that transformation RS 4 (4) generates quadratic transformation for the sixth Painlevé equation, in [5] a general notion of the RS-transformation for the so-called special functions of the isomonodromy type [6] were introduced and application to the construction of the algebraic solutions to the sixth Painlevé equation was discussed, in [7] we classified all RS 4 (3) transformations of the ranks ≤ 4.…”