论文标题
关于四阶二次均质差分方程的正式系列解决方案及其收敛性
On Formal Series Solutions To 4th-order Quadratic Homogeneous Differential Equations And Their Convergence
论文作者
论文摘要
众所周知，Painlevé方程的所有$τ$函数都满足四阶二次微分方程。在其中，对于III，V和VI方程，可以使用组合学明确表达形式的系列解决方案。在本文中，我们显示了正式系列的收敛性，包括更通用方程的解决方案。通过$τ$系列的绝对收敛性，由于它是$τ$系列的部分总和，因此同形块函数的收敛（$ c = 1 $）也随之而来。我们还表征了具有类似于Painlevé方程的Tau函数的串联解决方案的均匀二次方程的形式。 Painlevé方程分为六种类型，众所周知，可以通过从VI型转换为I类型来获得它们。
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 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 $τ$ series, the convergence of the conformal block function ($c=1$) also follows 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. The Painlevé equations are classified into six types, and it is known that they can be obtained by sequentially degenerating from type VI to type I.
