For an arbitrary solution to the KdV hierarchy, the generating series of logarithmic derivatives of the tau-function of the solution can be expressed by the basic matrix resolvent via algebraic manipulations. Based on this we develop in this paper two new formulae for the generating series by introducing a pair of wave functions of the solution. Applications to the Witten–Kontsevich tau-function, to the generalized Brézin–Gross–Witten (BGW) tau-function, as well as to a modular deformation of the generalized BGW tau-function which we call the Lamé tau-function are also given.

中文翻译：

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关于KdV层次结构的tau函数

对于KdV层次结构的任意解决方案，该解决方案的tau函数的对数导数的生成系列可以通过基本矩阵分解通过代数操作来表示。基于此，我们通过引入解的一对波动函数，在本文中为生成序列开发了两个新公式。还给出了Witten-Kontsevich tau函数，广义Brézin-Gross-Witten（BGW）tau函数以及广义BGW tau函数的模块化变形（我们称为Lamétau函数）的应用。 。