Inequalities for the dth residual crank moments of overpartitions

Abstract

Two analogues of the crank function are defined for overpartitions -the first residual crank and the second residual crank. This suggests an exploration of crank functions defined for overpartitions whose parts are divisible by an arbitrary d. We examine the positive moments of these crank functions while varying d and prove some inequalities.

Quasimodularity of the $k$th Residual Cranks

“…Work of Al-Saedi, Swisher, and the first author [ASMS0] extends the first and second residual cranks by defining the kth residual crank functions for all k > 1. To calculate the kth residual crank of the overpartition λ, take λ ′ to be the partition whose parts are the non-overlined parts of λ which vanish modulo k, divided by two.…”