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Expansions of Theta Functions and Applications

A. Raouf Chouikha
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Abstract

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

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A. Raouf Chouikha. 2006-05-04. Expansions of Theta Functions and Applications. https://arxiv.org/abs/math/0112137

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