Bilateral series in terms of mixed mock modular forms

Abstract

Abstract The number of strongly unimodal sequences of weight n is denoted by u ∗ ( n ) $u^{*}(n)$ . The generating functions for { u ∗ ( n ) } n = 1 ∞ $\{u^{*}(n)\}_{n=1}^{\infty}$ are U ∗ ( q ) = ∑ n = 1 ∞ u ∗ ( n ) q n $U^{*}(q)=\sum_{n=1}^{\infty}u^{*}(n)q^{n}$ . Rhoades recently gave a precise asymptotic for u ∗ ( n ) $u^{*}(n)$ by expressing U ∗ ( q ) $U^{*}(q)$ as a mixed mock modular form. In this note, by revisiting the mixed mock modular form associated to U ∗ ( q ) $U^{*}(q)$ , three new mixed mock modular forms are constructed by considering the bilateral series of U ∗ ( q ) $U^{*}(q)$ and the third order Ramanujan’s mock theta function f ( q ) $f(q)$ . 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 ) $B(f;q)$ of the third order mock theta function f ( q ) $f(q)$ is a mixed mock modular form.

Keywords

• bilateral series
• mixed mock modular form
• harmonic weak Maass form
• Appell-Lerch sums