Expansions of Theta Functions and Applications

Details Expansions of Theta Functions and Applications Expansions of Theta Functions and Applications

2001-12-13

arxiv.org/abs/math/0112137v4

Mathematics

Number Theory

23 pages

Scientific paper

We prove that the classical theta function $\theta_4$ may be expressed as $$ \theta_4(v,\tau) = \theta_4(0,\tau) \exp[- \sum_{p\geq 1} \sum_{k\geq 0} \frac {1}{p} \bigg(\frac {\sin \pi v}{(\sin (k+{1/2})\pi \tau)}\bigg)^{2p}].$$ 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

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