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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: Alexander Berkovich and Hamza Yesilyurt
Journal: Proc. Amer. Math. Soc. 142 (2014), 777-793
MSC (2010): Primary 11E16, 11E45, 11F03, 11P84
Published electronically: December 2, 2013
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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.


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Additional Information

Alexander Berkovich
Affiliation: Department of Mathematics, University of Florida, 358 Little Hall, Gainesville, Florida 32611
Email: alexb@ufl.edu

Hamza Yesilyurt
Affiliation: Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey
Email: hamza@fen.bilkent.edu.tr

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11816-2
PII: S 0002-9939(2013)11816-2
Keywords: Eta-quotients, binary quadratic forms, Rogers--Ramanujan functions, Ramanujan's lost notebook, Thompson series
Received by editor(s): April 7, 2012
Published electronically: December 2, 2013
Additional Notes: The first author’s research was partially supported by grant H98230-09-1-0051 of the National Security Agency
The second author’s research was partially supported by grant 109T669 from Tübitak
Communicated by: Ken Ono
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.