The Ramanujan identities under modular substitutions

Abstract

they appear in a form which suggests certain group-theoretical considerations, similar to those employed by Hecke in his theory of modular forms. In this way we transform the identities into new ones which are noteworthy because of the occurrence of the Legendre symbol and which, by a simple further argument, lead also to a proof of (1.1) and (1.2). An analogous identity for the modulus 13, given by Zuckerman, can be treated in the same way. G. N. Watson and H. S. Zuckerman have also derived identities for the moduli 52 and 72. These will lead us to certain modular equations, which in turn will shed some light on those identities.