Generalizations of Dyson's Rank on Overpartitions

We introduce a statistic on overpartitions called the $\overline{k}$-rank. When there are no overlined parts, this coincides with the $k$-rank of a partition introduced by Garvan. Moreover, it reduces to the D-rank of an overpartition when $k=2$. The generating function for the $\overline{k}$-rank of overpartitions is given. We also establish a relation between the generating function of self-3-conjugate overpartitions and the tenth order mock theta functions $X(q)$ and $\chi(q)$.