On the dual nature of partial theta functions and Appell-Lerch sums

摘要

In recent work, Hickerson and the author demonstrated that it is useful to think of Appell-Lerch sums as partial theta functions. This notion can be used to relate identities involving partial theta functions with identities involving Appell-Lerch sums. In this sense, Appell-Lerch sums and partial theta functions appear to be dual to each other. This duality theory is not unlike that found by Andrews between various sets of identities of Rogers-Ramanujan type with respect to Baxter's solution to the hard hexagon model of statistical mechanics. As an application we construct bilateral q-series with mixed mock modular behaviour. In subsequent work we see that our bilateral series are well-suited for computing radial limits of Ramanujan's mock theta functions. (C) 2014 Elsevier Inc. All rights reserved.