By introducing a dual notion between partial theta functions and Appell–Lerch sums, we find and prove a formula which expresses Hecke‐type double sums in terms of Appell–Lerch sums. Not only does our formula prove classical Hecke‐type sum identities such as those in work of Kac and Peterson on affine Lie Algebras and Hecke modular forms, but once we have the Hecke‐type forms for Ramanujan's mock theta functions our formula gives straightforward proofs of many of the classical mock theta identities. Indeed, we obtain a new proof of the mock theta conjectures. Our formula also applies to positive‐level string functions associated with admissible representations of the affine Lie Algebra A1(1) of Kac and Wakimoto.

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