Appell-Lerch sum

Etymology
First studied by Paul Émile Appell and Mathias Lerch.

Noun

 * 1)  A generalization of Lambert series with the form
 * $$\mu(u, v; \tau) = \frac{a^\frac{1}{2}}{\theta(v; \tau)}\sum_{n\in Z}\frac{(-b)^nq^{\frac{1}{2}n(n + 1)}}{1 - aq^n}$$

where
 * $$\displaystyle q = e^{2\pi i \tau},\quad a = e^{2\pi i u},\quad b = e^{2\pi i v}$$

and
 * $$\theta(v,\tau) = \sum_{n\in Z}(-1)^n b^{n + \frac{1}{2}}q^{\frac{1}{2}\left(n + \frac{1}{2}\right)^2}.$$