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In recent work, Hickerson and the author demonstrated that it is useful tothink of Appell--Lerch sums as partial theta functions. This notion can be usedto relate identities involving partial theta functions with identitiesinvolving Appell--Lerch sums. In this sense, Appell--Lerch sums and partialtheta functions appear to be dual to each other. This duality theory is notunlike that found by Andrews between various sets of identities ofRogers-Ramanujan type with respect to Baxter's solution to the hard hexagonmodel of statistical mechanics. As an application we construct bilateral$q$-series with mixed mock modular behaviour.

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