Infinite Series Identities, Transformations, and Evaluations

Abstract

Chapter 9 fully illustrates Hardy’s declaration in Ramanujan’s Collected Papers [15, p. xxxv], “It was his insight into algebraical formulae, transformations of infinite series, and so forth, that was most amazing.” This chapter has 35 sections containing 139 formulas of which many are, indeed, very beautiful and elegant. Ramanujan gives several transformations of power series leading to many striking series relations and attractive series evaluations. Most of Ramanujan’s initial efforts in this direction pertain to the dilogarithm and related functions. As is to be expected, these results are not new and can be traced back to Euler, Landen, Abel, and others. However, most of Ramanujan’s remaining findings on transformations of power series appear to be new.