On formal series solutions to 4th-order quadratic homogeneous differential equations and their convergence

Abstract It is known that all τ functions of the Painlevé equations satisfy the fourth-order quadratic differential equation. Among them, for the III, V, and VI equations, it is possible to express the formal series solutions combinatorially by using conformal blocks. In this paper, we show the convergence of the formal series, including the solutions of more general equations. The convergence of the conformal block function also follows in the case c = 1 by the absolute convergence of τ series, since it is a partial sum of the τ series. We also characterized the form of a homogeneous quadratic equation with a series solution similar to the tau functions of the Painlevé equations.