Appell-Lerch sum

First studied by Paul Émile Appell and Mathias Lerch.

Appell-Lerch sum (plural Appell-Lerch sums)

1. (mathematics) A generalization of Lambert series with the form

${\displaystyle \mu (u,v;\tau )={\frac {a^{\frac {1}{2}}}{\theta (v;\tau )}}\sum _{n\in Z}{\frac {(-b)^{n}q^{{\frac {1}{2}}n(n+1)}}{1-aq^{n}}}}$

where

${\displaystyle \displaystyle q=e^{2\pi i\tau },\quad a=e^{2\pi iu},\quad b=e^{2\pi iv}}$

and

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