We study a family of residual crank generating functions defined on overpartitions, the so-called [Formula: see text]th residual cranks. Specifically, the moment generating functions associated to these cranks exhibit quasimodularity properties which are dependent on the choice of [Formula: see text]. We also show that the second moments of these cranks admit a combinatoric interpretation as weighted overpartition counts. This interpretation gives a refinement to an existing inequality between crank moments of differing modulus [Formula: see text].

中文翻译：

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第 k 个剩余曲柄的准模性

我们研究了在过度分区上定义的一组剩余曲柄生成函数，即所谓的[公式：见文本]th剩余曲柄。具体来说，与这些曲柄相关的力矩生成函数表现出准模性质，这取决于 [公式：参见文本] 的选择。我们还表明，这些曲柄的二阶矩可以将组合解释为加权过度分区计数。这种解释对不同模数的曲柄力矩之间的现有不等式进行了改进[公式：见正文]。