Inequalities for the $d$th Residual Crank Moments of Overpartitions

TL;DR

This paper introduces two new crank functions for overpartitions and investigates their moments, establishing inequalities that deepen understanding of overpartition combinatorics related to divisibility by an arbitrary integer.

Contribution

It defines residual crank functions for overpartitions and proves inequalities for their moments across different divisibility parameters.

Findings

Defined two residual crank functions for overpartitions.

Proved inequalities for the positive moments of these crank functions.

Explored the behavior of moments as the divisibility parameter varies.

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.