A partition identity and the universal mock theta function $g_2$

Abstract

We prove analytic and combinatorial identities reminiscent of Schur's classical partition theorem. Specifically, we show that certain families of overpartitions whose parts satisfy gap conditions are equinumerous with partitions whose parts satisfy congruence conditions. Furthermore, if small parts are excluded, the resulting overpartitions are generated by the product of a modular form and Gordon and McIntosh's universal mock theta function. Finally, we give an interpretation for the universal mock theta function at real arguments in terms of certain conditional probabilities.