RAMANUJANS THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES

In his famous paper on modular equations and approximations to pi, Ram anujan offers several series representations for 1/pi, which he claims are derived from ''corresponding theories'' in which the classical ba se q is replaced by one of three other bases. The formulas for 1/pi we re only recently proved by J. M. and P. B. Borwein in 1987, but these ''corresponding theories'' have never been heretofore developed. Howev er, on six pages of his notebooks, Ramanujan gives approximately 50 re sults without proofs in these theories. The purpose of this paper is t o prove all of these claims, and several further results are establish ed as well.