On conjectures of Koike and Somos for modular identities for the Rogers–Ramanujan functions

Abstract

In this paper, we prove modular identities conjectured by Koike and Somos for the Rogers–Ramanujan functions. Our methods focus on approaches that Ramanujan could have employed, including the theory of modular equations, dissections of identities, and methods derived from the approaches of Watson and Rogers.