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Ramanujan’s explicit values for the classical theta-function. (English) Zbl 0862.33016
In Ramanujan’s notation the theta function in question is $$\varphi(q) = \sum^\infty_{n= -\infty} q^{n^2}$$, where $$|q|<1$$. In his notebooks, Ramanujan recorded values of $$\varphi(e^{-n \pi})/ \varphi (e^{-\pi})$$ for $$n=3, 7, 9$$ and 45. Since $$\varphi(e^{-\pi}) = \pi^{1/4}/ \Gamma(3/4)$$ this gives $$\varphi (e^{-n \pi})$$ explicitly in closed form. This paper gives proofs for these evaluations using modular equations. The same method also evaluates $$\varphi (e^{-n \pi})$$ for $$n=13, 27$$ and 63, providing new results not claimed by Ramanujan.

##### MSC:
 11F27 Theta series; Weil representation; theta correspondences 11F20 Dedekind eta function, Dedekind sums 11A55 Continued fractions
##### Keywords:
theta function; modular equations
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##### References:
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