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

It is known that all tau functions of the Painlev & eacute; 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 tau series, since it is a partial sum of the tau series. We also characterized the form of a homogeneous quadratic equation with a series solution similar to the tau functions of the Painlev & eacute; equations.