Quasimodularity of the kth residual cranks

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].