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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:
 [1] DOI: 10.1017/S0305004100053731 · Zbl 0356.33003 [2] DOI: 10.1016/S0747-7171(85)80014-6 · Zbl 0574.12002 [3] Whittaker, A Course of Modern Analysis (1966) [4] DOI: 10.1112/plms/s2-42.1.377 · Zbl 0015.38903 [5] DOI: 10.1093/qmath/os-3.1.189 · Zbl 0005.29604 [6] Ramanujan, Quart. J. Math 45 pp 350– (1914) [7] Borwein, Pi and the AGM (1987) [8] Hall, Higher Algebra (1957) [9] DOI: 10.1017/S0305004100069723 · Zbl 0732.11030 [10] Cox, Primes of the form x (1989) [11] DOI: 10.1093/imanum/12.4.519 · Zbl 0758.65008 [12] DOI: 10.2307/2001551 · Zbl 0725.33014 [13] DOI: 10.1007/BF03024284 · Zbl 0816.11069
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