Hypergeometric Series, I

Abstract

In 1923, Hardy published a paper [1],[7, pp. 505–516] providing an overview of the contents of Chapter 12 of the first notebook. This chapter which corresponds to Chapter 10 of the second notebook, is connected primarily with hypergeometric series. It should be emphasized that Hardy gave only a brief survey of Chapter 12; this chapter contains many interesting results not mentioned by Hardy, and Chapter 10 of the second notebook possesses material not found in the first. Quite remarkably, Ramanujan independently discovered a great number of the primary classical theorems in the theory of hypergeometric series. In Particular, he rediscovered well-known theorems of Gauss, Kummer, Dougall, Dixon, Saalschütz, and Thomae, as well as special cases of Whipple’s transformation. Unfortunately, Ramanujan left us little knowledge as to how he made his beautiful discoveries about hypergeometric series. The first notebook is found after Entry 8, which is Gauss’s theorem. We shall present this argument of Ramanujan in the sequel.