论文标题
关于超几何方程的tau函数
On the tau function of the hypergeometric equation
论文作者
论文摘要
Kummer公式用于高斯的超几个体系功能，在经典上解决了具有三个紫红色奇异性的微分方程的等级方程组的单形图。我们使用最近提出的想法来定义诸如扩展单构型符号切除型的生成函数的tau功能。这种公式使我们能够确定tau功能对单肌数据的依赖性。使用单莫氏菌问题的明确解决方案，然后用barnes $ g $功能明确编写tau功能。 特别是，如果将紫红色的奇异性放在$ 0 $，$ 1 $和$ \ infty $中，则可以提供Iorgov-Gamayun-Lisovyy渐近公式的结构常数，用于painlevéviVi方程的解决方案。
The monodromy map for a rank-two system of differential equations with three Fuchsian singularities is classically solved by the Kummer formulæ for Gauss' hypergeometric functions. We define the tau-function of such a system as the generating function of the extended monodromy symplectomorphism, using an idea recently developed. This formulation allows us to determine the dependence of the tau-function on the monodromy data. Using the explicit solution of the monodromy problem, the tau-function is then explicitly written in terms of Barnes $G$-function. In particular, if the Fuchsian singularities are placed to $0$, $1$ and $\infty$, this gives the structure constants of the asymptotical formula of Iorgov-Gamayun-Lisovyy for solutions of Painlevé VI equation.
