On a generalized crank for k-colored partitions

A generalized crank (k-crank) for k-colored partitions is introduced. Following the work of Andrews-Lewis and Ji-Zhao, we derive two results for this newly defined k-crank. Namely, we first obtain some inequalities between the k-crank counts Mk(r,m,n) for m=2,3 and 4, then we prove the positivity of symmetrized even k-crank moments weighted by the parity for k=2 and 3. We conclude with several remarks on furthering the study initiated here.