Appell-Lerch sum

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English

English Wikipedia has an article on:

Mock modular form

Etymology

First studied by Paul Émile Appell and Mathias Lerch.

Noun

Appell-Lerch sum (plural Appell-Lerch sums)

(mathematics) 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-aq^{n}}}

where

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

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}}.

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