We introduce a new class of combinatorially defined rational functions and apply them to deduce explicit formulae for local ideal zeta functions associated to the members of a large class of nilpotent Lie rings which contains the free class-2-nilpotent Lie rings and is stable under direct products. Our results unify and generalize a substantial number of previous computations. We show that the new rational functions, and thus also the local zeta functions under consideration, enjoy a self-reciprocity property, expressed in terms of a functional equation upon inversion of variables. We establish a conjecture of Grunewald, Segal, and Smith on the uniformity of normal zeta functions of finitely generated free class-2-nilpotent groups.

我们引入了一类新的组合定义的有理函数，并应用它们推导出与一大类幂零李环的成员相关的局部理想 zeta 函数的显式公式，该幂零李环包含自由的 2 类幂零李环，并且在直接条件下是稳定的产品。我们的结果统一并概括了以前的大量计算。我们表明，新的有理函数以及所考虑的局部 zeta 函数具有自互易性质，以变量反演时的函数方程的形式表示。我们建立了 Grunewald、Segal 和 Smith 关于有限生成的自由 2 类幂零群的正常 zeta 函数的一致性的猜想。