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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
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