Buffered Frobenius representations of overpartitions

Overpartitions arise as a way to give combinatorial interpretation of the coefficients of $q$-hypergeometric series. We generalize the first and second Frobenius representations of overpartitions into two infinite families, which in turn correspond to generalizations of the Dyson rank and $M_2$-rank of overpartitions, the $M_d$-ranks. The moments of these new rank functions give rise to mock modular forms, as is the case with the Dyson and $M_2$-ranks.