On a new circular summation of theta functions
type: Item
instance of: scholarly article
title: On a new circular summation of theta functions
zbMATH Open document ID: 1196.33016
author: Song Heng Chan
author: Zhi-Guo Liu
published in: Journal of Number Theory
publication date: 2010-05-17T00:00:00Z
review text: An example of circular summation of theta functions is the formula \[ \sum^{n-1}_{k=0} \theta_3\Biggl(z+{k\pi\over n}, \tau\Biggr)= n\theta_3(nz, n\tau), \] proved by \textit{M. Boon}, \textit{M. L. Glasser}, \textit{J. Zak} and \textit{I. J. Zucker} [J. Phys. A 15, 3439--3440 (1982; Zbl 0543.33004)], where \[ \theta_3(z,\tau) =\sum^\infty_{n=-\infty} q^{n^2/2} e^{2niz} \] with \(q= e^{2\pi ir}\), \(\text{Im}(z)>0\) is Jacobi's theta function. The first instance of circular summation is known from Ramanujan's Lost Notebook. The authors prove formulas which comprise as special cases most of the previously known formulas on circular summation. One of them reads \[ \sum^{mn-1}_{k=0}\;\pi^n_{j=1} \theta_3\Biggl(z+ y_j+ {k\pi\over mn},\tau\Biggr)= G_{m,n}(y_1,\dots, y_n,\tau) \theta_3(mnz, m^2n\tau), \] where \(y_1,\dots, y_n\in\mathbb{C}\), \(y_1+\cdots+ y_n= 0\) and \(G_{m,n}\) is explicitly given in terms of an infinite series independent from \(z\). One among several corollaries is \[ {\eta^3(\tau)\over \eta(3\tau)}= \sum_{x,y\in\mathbb{Z}} q^{x^2+ xy+ y^2} \omega^{x-y} \] with \(\omega= e^{2\pi i/3}\), relating a quotient of Dedekind's eta function \(\eta(\tau)\) to the ring of Eisenstein integers.
reviewed by: Günter Köhler
Mathematics Subject Classification ID: 33E05
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F20
Mathematics Subject Classification ID: 11F27
zbMATH DE Number: 5708637
zbMATH Keywords: elliptic function
zbMATH Keywords: theta function
zbMATH Keywords: circular summation
zbMATH Keywords: addition formula
zbMATH Keywords: modular equation
MaRDI profile type: Publication
full work available at URL: j.jnt.2009.09.011
OpenAlex ID: W1968413824
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DOI: 10.1016/J.JNT.2009.09.011
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description: scientific article; zbMATH DE number 5708637