Appell-Lerch sum

English

Etymology

First studied by Paul Émile Appell and Mathias Lerch.

Noun

1. Template:Lb A generalization of Lambert series with the form

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

where

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

and

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