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

TL;DR

This paper explores the connections between Rogers-Ramanujan functions, eta-quotients, and binary quadratic forms, revealing new identities through Hecke actions and extending Ramanujan's original identities.

Contribution

It introduces a novel approach using Hecke operators on eta products to derive new identities for Rogers-Ramanujan functions and binary quadratic forms.

Findings

New identities for Rogers-Ramanujan functions derived

Hecke actions relate eta-quotients to quadratic forms

Extended Ramanujan's original identities

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 functions that appear 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 turn to find identities involving binary quadratic forms.