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

Abstract

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.

To be published in Advances in Mathematics

References in corpus (4)

• Quantum Black Holes, Wall Crossing, and Mock Modular Forms
• Hecke-type double sums, Appell-Lerch sums, and mock theta functions (I)
• On three third order mock theta functions and Hecke-type double sums
• Ramanujan's radial limits and mixed mock modular bilateral $q$-hypergeometric series

Cited by in corpus (13)

• Dyson's Ranks and Appell-Lerch Sums
• Ramanujan's radial limits and mixed mock modular bilateral $q$-hypergeometric series
• A double-sum Kronecker-type identity
• Renormalization and quantum modular forms, part II: Mock theta functions
• Proofs of Some Conjectures of Chan on Appell-Lerch Sums
• The mixed mock modularity of certain duals of generalized quantum modular forms of Hikami and Lovejoy
• Eulerian series as modular forms revisited
• Mock Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series
• Orientation Reversal and the Chern-Simons Natural Boundary
• A general formula for Hecke-type false theta functions
• Asymptotics of generalized partial theta functions with a Dirichlet character
• Properties of Lerch Sums and Ramanujan's Mock Theta Functions
• Generating Functions of the Hurwitz Class Numbers Associated with Certain Mock Theta Functions