Asymptotics for $k$-crank of $k$-colored partitions

In this paper, we obtain asymptotic formulas for $k$-crank of $k$-colored partitions. Let $M_k(a,c;n)$ denote the number of $k$-colored partitions of $n$ with a $k$-crank congruent to $a$ mod $c$. For the cases $k=2,3,4$, Fu and Tang derived several inequality relations for $M_k(a,c;n)$ using generating functions. We employ the Hardy-Ramanujan Circle Method to extend the results of Fu and Tang. Furthermore, additional inequality relations for $M_k(a,c;n)$ have been established, such as logarithmic concavity and logarithmic subadditivity.