On Formal Series Solutions To 4th-order Quadratic Homogeneous Differential Equations And Their Convergence

It is known that all $\tau$ functions of the Painlev\'{e} 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 explicitly by using combinatorics. In this paper, we show the convergence of the formal series, including the solutions of more general equations. And by the absolute convergence of $\tau$ series, the convergence of the conformal block function ($c=1$) also follows 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\'{e} equations. The Painlev\'{e} equations are classified into six types, and it is known that they can be obtained by sequentially degenerating from type VI to type I.