论文标题
从Heun类方程到Painlevé方程
From Heun Class Equations to Painlevé Equations
论文作者
论文摘要
在论文的第一部分中，我们讨论了复杂域中的线性二阶差分方程，尤其是Heun类方程，即Heun方程及其汇合案例。我们论文的第二部分致力于PainlevéI-VI方程。我们的理念是以统一的方式对待这些方程式。这种哲学对于Heun班级方程式特别有效。我们将其分类为5种超构型，分为10种类型（不计算琐碎的情况）。我们还以统一的变形HEUN类方程进行介绍，其中包含其他非质量奇异性。我们表明，变形的Heun类方程与所有Painlevé方程之间存在直接关系。特别是，Painlevé方程也可以分为5种超级型，并细分为10种类型。这种关系并不容易以一种完全统一的方式描述，因为“时间变量”的选择可能取决于类型。我们描述了几种可能的“时间变量”的统一处理。
In the first part of our paper we discuss linear 2nd order differential equations in the complex domain, especially Heun class equations, that is, the Heun equation and its confluent cases. The second part of our paper is devoted to Painlevé I-VI equations. Our philosophy is to treat these families of equations in a unified way. This philosophy works especially well for Heun class equations. We discuss its classification into 5 supertypes, subdivided into 10 types (not counting trivial cases). We also introduce in a unified way deformed Heun class equations, which contain an additional nonlogarithmic singularity. We show that there is a direct relationship between deformed Heun class equations and all Painlevé equations. In particular, Painlevé equations can be also divided into 5 supertypes, and subdivided into 10 types. This relationship is not so easy to describe in a completely unified way, because the choice of the ''time variable'' may depend on the type. We describe unified treatments for several possible ''time variables''.
