Solutions of Darboux Equations, its Degeneration and Painlevé VI Equations

Abstract

In this paper, we study the Darboux equations in both classical and system form, which give the elliptic Painlevé VI equations by the isomonodromy deformation method. Then we establish the full correspondence between the special Darboux equations and the special Painlevé VI equations. Instead of the system form, we especially focus on the Darboux equation in a scalar form, which is the generalization of the classical Lamé equation. We introduce a new infinite series expansion (in terms of the compositions of hypergeometric functions and Jacobi elliptic functions) for the solutions of the Darboux equations and regard special solutions of the Darboux equations as those terminating series. The Darboux equations characterized in this manner have an almost (but not completely) full correspondence to the special types of the Painlevé VI equations. Finally, we discuss the convergence of these infinite series expansions. Contents 1 Introduction 1 2 The Darboux Connection and the Elliptic Painlevé VI Equation 4 2.

Invariant subspaces of biconfluent Heun operators and special solutions of Painlevé IV

“…Indeed special function expansions similar to (1.12) for Fuchsian type (scalar) differential equations appeared in earlier works of Heine [19] for the Lamé equation, and Kimura [25], Erdelyi [11], Wolfrat et al [44] for the Heun equations. We refer the reader to [4] for a correspondence between special solutions of the Darboux equation (which is an elliptic version of the Heun equation) and special solutions of Painlevé VI.…”