Solutions of Darboux Equations, its Degeneration and Painlev\'e VI Equations

In this paper, we study the Darboux equations in both classical and system form, which give the elliptic Painlev\'e VI equations by the isomonodromy deformation method. Then we establish the full correspondence between the special Darboux equations and the special Painlev\'e 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 Lam\'{e} equation. We introduce a new infinite series expansion (in terms of the compositions of hypergeometric functions and Jacobi elliptic functions) %around each of the four regular singular points of the 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 Painlev\'e VI equations. Finally, we discuss the convergence of these infinite series expansions.