Mock and mixed mock modular forms in the lower half-plane

We study mock and mixed mock modular forms in the lower half-plane. In particular, our results apply to Zwegers’ three-variable mock Jacobi form μ(u, v; τ), three-variable generalizations of the universal mock modular partition rank generating function, and the quantum and mock modular strongly unimodal sequence rank generating function. We do not rely upon the analytic properties of these functions; we establish our results concisely using the theory of q-hypergeometric series and partial theta functions. We extend related results of Ramanujan, Hikami, and prior work of the author with Bringmann and Rhoades, and also incorporate more recent aspects of the theory pertaining to quantum modular forms and the behavior of these functions at rational numbers when viewed as functions of τ (or equivalently, at roots of unity when viewed as functions of q). Mathematics Subject Classification. 11F37, 33D15, 11F27.