Ramanujan’s theories of elliptic functions to alternative bases

Abstract

In his famous paper on modular equations and approximations to π \pi , Ramanujan offers several series representations for 1 / π 1/\pi , which he claims are derived from "corresponding theories" in which the classical base q q is replaced by one of three other bases. The formulas for 1 / π 1/\pi were only recently proved by J. M. and P. B. Borwein in 1987, but these "corresponding theories" have never been heretofore developed. However, on six pages of his notebooks, Ramanujan gives approximately 50 results without proofs in these theories. The purpose of this paper is to prove all of these claims, and several further results are established as well.