Appell-Lerch sums

introduction

one way to construct mock theta functions
characters of representations in (nonrational) conformal field theory models based on Lie superalgebras

Appell-Lerch sum

Appell–Lerch sums were first studied by Paul Émile Appell (1884) and Mathias Lerch (1892). Watson studied the order 3 mock theta functions by expressing them in terms of Appell–Lerch sums, and Zwegers used them to show that mock theta functions are essentially mock modular forms.

The Appell–Lerch series is

\(\mu(u,v;\tau) = \frac{a^{1/2}}{\theta(v;\tau)}\sum_{n\in Z}\frac{(-b)^nq^{n(n+1)/2}}{1-aq^n}\)

where

\(\displaystyle q= e^{2\pi i \tau},\quad a= e^{2\pi i u},\quad b= e^{2\pi i v}\)

and

\(\theta(v,\tau) = \sum_{n\in Z}(-1)^n b^{n+1/2}q^{(n+1/2)^2/2}\)

The modified series

\[\hat\mu(u,v;\tau) = \mu(u,v;\tau)-R(u-v;\tau)/2\]

where

\[R(z;\tau) = \sum_{\nu\in Z+1/2}(-1)^{\nu-1/2}({\rm sign}(\nu)-E((\nu+\Im(z)/y)\sqrt{2y}))e^{-2\pi i \nu z}q^{-\nu^2/2}\]

and y = Im(τ) and

\[E(z) = 2\int_0^ze^{-\pi u^2}\,du\]

satisfies the following transformation properties

\[\hat\mu(u+1,v;\tau) = a^{-1}bq^{-1/2}\hat\mu(u+\tau,v;\tau) = -\hat\mu(u,v;\tau),\]

\[e^{2\pi i/8}\hat\mu(u,v;\tau+1) = \hat\mu(u,v;\tau) = -(\tau/i)^{-1/2}e^{\pi i (u-v)^2/\tau}\hat\mu(u/\tau,v/\tau;-1/\tau).\]

In other words the modified Appell–Lerch series transforms like a modular form with respect to τ. Since mock theta functions can be expressed in terms of Appell–Lerch series this means that mock theta functions transform like modular forms if they have a certain non-analytic series added to them.

higher level Appell function

higher-level Appell functions
- a particular instance of indefinite theta series

history

http://www.google.com/search?hl=en&tbs=tl:1&q=

related items

encyclopedia

http://ko.wikipedia.org/wiki/
http://en.wikipedia.org/wiki/Appell
http://en.wikipedia.org/wiki/
http://en.wikipedia.org/wiki/
Princeton companion to mathematics(Companion_to_Mathematics.pdf)

question and answers(Math Overflow)