On the dual nature theory of bilateral series associated to mock theta functions

Abstract

In recent work, Hickerson and Mortenson introduced a dual notion between Appell–Lerch sums and partial theta functions. In this sense, Appell–Lerch sums and partial theta functions appear to be dual to each other. In this paper, by making the substitution [Formula: see text] in the tail of the associated bilateral series of mock theta functions and universal mock theta functions, we demonstrate how to obtain duals of the second type in terms of Appell–Lerch sums defined by Mortenson for such functions. Then by using the substitution [Formula: see text] in duals of the second type of each bilateral series, we present how to translate between identities expressing [Formula: see text]-hypergeometric series in terms of Appell–Lerch sums and identities expressing [Formula: see text]-hypergeometric series in terms of partial theta functions. Indeed, we obtain only four duals in terms of partial theta functions of duals of the second type in terms of Appell–Lerch sums of bilateral series associated to mock theta functions. As an application, we construct Ramanujan radial limits by using these bilateral series with mock modular behavior in terms of Appell–Lerch sums for some order mock theta functions. This method is well-suited for the other order mock theta functions.