Ramanujan described a mock theta function as a \(q\)-series \(f(q)\) satisfying the following:
(1) For every root of unity \(\zeta\), there exists a theta function (i.e., a modular form) \(\theta_{\zeta}(q)\) such that \(f(q) - \theta_{\zeta}(q)\) is bounded as \(q \to \zeta\) radially.
(2) There is no theta function which works for all \(\zeta\).
In this paper the authors discuss how mock theta identities imply (1) for the fifth order mock theta functions and most of the sixth and eighth order mock theta functions. For example, consider the fifth order mock theta functions \[ f_0(q) := \sum_{n \geq 0} \frac{q^{n^2}}{(-q;q)_n} \] and \[ \psi_0(q) := \sum_{n \geq 0} q^{\binom{n+2}{2}}(-q;q)_n. \] Identities of Ramanujan give that \[ f_0(q) + 2\psi_0(q) = B(q), \] where \[ B(q) := \frac{(q)_{\infty}}{(-q)_{\infty}(q;q^5)_{\infty}(q^4;q^5)_{\infty}} + 4q\frac{(q^4;q^4)_{\infty}(-q^2;q^2)_{\infty}}{(q^8;q^{20})_{\infty}(q^{12};q^{20})_{\infty}} \] is essentially a modular form of weight \(1/2\). Hence if \(\zeta\) is, for example, a primitive \(2k\)th root of unity, then we have \[ \lim_{q \to \zeta}(f_0(q) - B(q)) = -2 \sum_{n=0}^{k-1}\zeta^{\binom{n+2}{2}}(-\zeta;\zeta)_n, \] when \(q \to \zeta\) radially.
The “bilateral” of the paper’s title refers to the fact that the sums like \(f_0(q) + 2\psi_0(q)\) may be regarded as bilateral series.