Quantum q-series identities

As analytic statements, classical q-series identities are equalities between power series for |q|<1. This paper concerns a different kind of identity, which we call a quantum q-series identity. By a quantum q-series identity we mean an identity which does not hold as an equality between power series inside the unit disk in the classical sense, but does hold on a dense subset of the boundary -- namely, at roots of unity. Prototypical examples were given over thirty years ago by Cohen and more recently by Bryson-Ono-Pitman-Rhoades and Folsom-Ki-Vu-Yang. We show how these and numerous other quantum q-series identities can all be easily deduced from one simple classical q-series transformation. We then use other results from the theory of q-hypergeometric series to find many more such identities. Some of these involve Ramanujan's false theta functions and/or mock theta functions.