On Rogers-Ramanujan functions, binary quadratic froms and eta-quotients

Abstract

In a handwritten manuscript published with his lost notebook, Ramanujan stated without proofs forty identities for the Rogers-Ramanujan functions. We observe that the function that appears in Ramanujan's identities can be obtained from a Hecke action on a certain family of eta products. We establish further Hecke-type relations for these functions involving binary quadratic forms. Our observations enable us to find new identities for the Rogers-Ramanujan functions and also to use such identities in return to find identities involving binary quadratic forms. © 2013 American Mathematical Society.