Based on the matrix-resolvent approach, for an arbitrary solution to the discrete KdV hierarchy, we define the tau-function of the solution, and compare it with another tau-function of the solution defined via reduction of the Toda lattice hierarchy. Explicit formulae for generating series of logarithmic derivatives of the tau-functions are obtained, and applications to enumeration of ribbon graphs with even valencies and to certain special cubic Hodge integrals are considered.

中文翻译：

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矩阵解析和离散 KdV 层次结构

基于矩阵解析方法，对于离散 KdV 层次结构的任意解，我们定义解的 tau 函数，并将其与通过简化 Toda 点阵层次定义的解的另一个 tau 函数进行比较。获得了生成 tau 函数的对数导数序列的显式公式，并考虑了在具有偶数化合价的带状图和某些特殊三次霍奇积分中的应用。