On the modularity of the odd rank generating functions

Abstract

To provide partition-theoretic interpretations to Watson's the third-order mock theta function $\ddot{I}‰(q)$, Andrews defined the odd Durfee symbols and odd ranks. Motivated by Andrews' work, arithmetic properties of odd ranks are widely studied recently. In this paper, we obtain transformation formulas of the odd rank generating functions, which are used to construct families of weak Maass forms and weakly modular forms. As an application,we provide explicit identities for odd ranks modulo 5, analogous to Ramanujan's classical partition identities.