Duke Mathematical Journal

A framework of Rogers–Ramanujan identities and their arithmetic properties

Michael J. Griffin, Ken Ono, and S. Ole Warnaar

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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.

Article information

Duke Math. J., Volume 165, Number 8 (2016), 1475-1527.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G16: Elliptic and modular units [See also 11R27]
Secondary: 05E05: Symmetric functions and generalizations 05E10: Combinatorial aspects of representation theory [See also 20C30] 11P84: Partition identities; identities of Rogers-Ramanujan type 17B67: Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 33D67: Basic hypergeometric functions associated with root systems

Rogers–Ramanujan identities modular units


Griffin, Michael J.; Ono, Ken; Warnaar, S. Ole. A framework of Rogers–Ramanujan identities and their arithmetic properties. Duke Math. J. 165 (2016), no. 8, 1475--1527. doi:10.1215/00127094-3449994. https://projecteuclid.org/euclid.dmj/1454683424

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