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

2012/04/04 by Alexander Berkovich, Berkovich, Alexander, Hamza Yesilyurt +1
Mathematics · #11E16 #11E45 #11F03 #11P84 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11E16 #msc:11E45 #msc:11F03 #msc:11P84

paper · pdf · doi:10.48550/arxiv.1204.1092

14 pages, no figures, no typos. To appear in Proceedings of the AMS

arxiv created 2012/07/22 · arxiv updated 2012/07/24

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.

Related

• Quadratic Forms, Exact Covering Systems, and Product Identities for Theta Functions
• The Rogers-Ramanujan Identities and Cauchy's Identity
• The Rogers-Ramanujan dissection of a theta function
• Ramanujan's Identities and Representation of Integers by Certain Binary and Quaternary Quadratic Forms
• The Rogers--Ramanujan continued fraction