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Ramanujan’s rational and algebraic series for $$1/\pi$$. (English) Zbl 0699.10044
From the introduction: S. Ramanujan [Q. J. Math., Oxf. 45, 350–372 (1914; JFM 45.1249.01)] sketched the genesis of 3 remarkable series for $$1/\pi$$ and with essentially no explanation, he gave 14 more remarkable series. Hardy, quoting Mordell, observes that “it is unfortunate that Ramanujan has not developed in detail the corresponding theories”. In this paper we construct various general classes of hypergeometric-like power series for $$1/\pi$$, and for several related quantities. In each case the power is a well-known invariant from elliptic function theory and the coefficients involve a similar invariant. In particular, we recover all but 2 of Ramanujan’s series and largely explain Ramanujan’s “corresponding theories”. A complete treatment appeared in the authors’ book [Pi and the AGM. New York: Wiley (1987; Zbl 0611.10001)] which we follow closely in the development of the material and which explains the two missing series.

MSC:
 11Y60 Evaluation of number-theoretic constants 33C20 Generalized hypergeometric series, $${}_pF_q$$ 11F27 Theta series; Weil representation; theta correspondences