Translator Disclaimer
1 June 2016 A framework of Rogers–Ramanujan identities and their arithmetic properties
Michael J. Griffin, Ken Ono, S. Ole Warnaar
Duke Math. J. 165(8): 1475-1527 (1 June 2016). DOI: 10.1215/00127094-3449994

Abstract

The two Rogers–Ramanujan q-series

n=0qn(n+σ)(1q)(1qn), where σ=0,1, play many roles in mathematics and physics. By the Rogers–Ramanujan identities, they are essentially modular functions. Their quotient, the Rogers–Ramanujan continued fraction, has the special property that its singular values are algebraic integral units. We find a framework which extends the Rogers–Ramanujan identities to doubly infinite families of q-series identities. If a{1,2} and m,n1, then we have

λ,λ1mqa|λ|P2λ(1,q,q2,;qn)=[infiniteproductmodularfunction],where the Pλ(x1,x2,;q) are Hall–Littlewood polynomials. These q-series are specialized characters of affine Kac–Moody algebras. Generalizing the Rogers–Ramanujan continued fraction, we prove in the case of A2n(2) that the relevant q-series quotients are integral units.

Citation

Download Citation

Michael J. Griffin. Ken Ono. S. Ole Warnaar. "A framework of Rogers–Ramanujan identities and their arithmetic properties." Duke Math. J. 165 (8) 1475 - 1527, 1 June 2016. https://doi.org/10.1215/00127094-3449994

Information

Received: 1 May 2014; Revised: 18 May 2015; Published: 1 June 2016
First available in Project Euclid: 5 February 2016

zbMATH: 06603536
MathSciNet: MR3504177
Digital Object Identifier: 10.1215/00127094-3449994

Subjects:
Primary: 11G16
Secondary: 05E05, 05E10, 11P84, 17B67, 33D67

Rights: Copyright © 2016 Duke University Press

JOURNAL ARTICLE
53 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.165 • No. 8 • 1 June 2016
Back to Top