Bilateral series in terms of mixed mock modular forms

Abstract

The number of strongly unimodal sequences of weight n is denoted by $u^{*}(n)$ . The generating functions for $\{u^{*}(n)\}_{n=1}^{\infty}$ are $U^{*}(q)=\sum_{n=1}^{\infty}u^{*}(n)q^{n}$ . Rhoades recently gave a precise asymptotic for $u^{*}(n)$ by expressing $U^{*}(q)$ as a mixed mock modular form. In this note, by revisiting the mixed mock modular form associated to $U^{*}(q)$ , three new mixed mock modular forms are constructed by considering the bilateral series of $U^{*}(q)$ and the third order Ramanujan’s mock theta function $f(q)$ . The inner relationships among them are discussed although they are defined in different ways. These new mixed mock modular forms can be expressed in terms of Appell-Lerch sums. The related mock theta functions can be completed as harmonic weak Maass forms. As an application, we give a proof for the claim by Bajpai et al. that the bilateral series $B(f;q)$ of the third order mock theta function $f(q)$ is a mixed mock modular form.