Generalizations of Dyson’s rank on overpartitions

Abstract

We introduce a statistic on overpartitions called the [Formula: see text]-rank. When there are no overlined parts, this coincides with the [Formula: see text]-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 [Formula: see text]-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 [Formula: see text].