"Appell-Lerch sums"의 두 판 사이의 차이

2010년 4월 7일 (수) 08:33 판

introduction

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

history

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

related items

Kac-Wakimoto modules

books

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)

blogs

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articles

Some characters of Kac and Wakimoto and nonholomorphic modular functions.
- K. Bringmann and K. Ono,
Math. Annalen 345, pages 547-558 (2009)
Higher String Functions, Higher-Level Appell Functions, and the Logarithmic sℓ︿2k/u(1) CFT Model
- A. M. Semikhatov
Higher-Level Appell Functions, Modular Transformations, and Characters
- A.M. Semikhatov

Mock Theta Functions
- Sander Zwegers, 2002
Integrable highest weight modules over affine superalgebras and Appell’s function
- Kac V.G., Wakimoto M, Commun. Math. Phys. 215(3), 631–682 (2001)

experts on the field

http://arxiv.org/

TeX

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@@ 139번째 줄: / 139번째 줄: @@
 <h5>articles</h5>
-* [http://dx.doi.org/10.1007/s002200000315 Integrable highest weight modules over affine superalgebras and Appell’s function]<br>
+*  Some characters of Kac and Wakimoto and nonholomorphic modular functions.<br>
-** Kac V.G., Wakimoto M, Commun. Math. Phys. '''215'''(3), 631–682 (2001)<br>
+** K. Bringmann and K. Ono,
+* Math. Annalen 345, pages 547-558 (2009)
+*  Higher String Functions, Higher-Level Appell Functions, and the Logarithmic sℓ︿2k/u(1)  CFT Model<br>
+** A. M. Semikhatov
+*  Higher-Level Appell Functions, Modular Transformations, and Characters<br>
+** A.M. Semikhatov
 * [http://front.math.ucdavis.edu/0807.4834 Mock Theta Functions]<br>
 ** Sander [http://front.math.ucdavis.edu/author/S.Zwegers Zwegers], 2002
+* [http://dx.doi.org/10.1007/s002200000315 Integrable highest weight modules over affine superalgebras and Appell’s function]<br>
+** Kac V.G., Wakimoto M, Commun. Math. Phys. '''215'''(3), 631–682 (2001)<br>
 * [[2010년 books and articles|논문정리]]