arXivDaily arXiv每日学术速递 周一至周五更新
arXiv周末暂无论文更新，休息一下吧，周末愉快~~
arXiv math/0112137math.NTmath.CA
收藏

θ函数的展开式及其应用

Expansions of Theta Functions and Applications

A. Raouf Chouikha

备注：
23 pages
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

我们证明了经典的θ函数$θ_4$可以表示为$$ θ_4(v,τ) = θ_4(0,τ) \exp[- \sum_{p\geq 1} \sum_{k\geq 0} \frac {1}{p} \bigg(\frac {\sin πv}{(\sin (k+{1/2})πτ)}\bigg)^{2p}].$$。我们得到了其他三个θ函数的类似展开式，因为它们是相关的。这些结果有几个后果。特别地，将导出Weierstrass椭圆函数的展开式。模群的作用和其他算术性质也将被考虑。最后，利用Rogers-Ramanujan连分数的一个新表达式，给出了Rogers恒等式的一个简单证明. { \it关键词和短语}：θ函数，椭圆函数，q-级数，傅里叶级数，连分数

英文摘要：

We prove that the classical theta function $θ_4$ may be expressed as $$ θ_4(v,τ) = θ_4(0,τ) \exp[- \sum_{p\geq 1} \sum_{k\geq 0} \frac {1}{p} \bigg(\frac {\sin πv}{(\sin (k+{1/2})πτ)}\bigg)^{2p}].$$ We obtain an analogous expansion for the three other theta functions since they are related. \\ These results have several consequences. In particular, an expansion of the Weierstrass elliptic function will be derived. Actions of the modular group and other arithmetical properties will also be considered. Finally using a new expression for the Rogers-Ramanujan continued fraction we produce a simple proof of a Rogers identity. {\it Key words and phrases} : theta functions, elliptic functions, q-series, Fourier series, continued fractions

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑