A framework of Rogers–Ramanujan identities and their arithmetic properties

By Michael J. Griffin, Ken Ichi Ono. 2014

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The two Rogers-Ramanujan q-series 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, n ≥ 1, then we have [infinite product modular function], where the P λ(x, x, ...; 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 A that the relevant q-series quotients are integral units.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0d6ff0c5f2937b23033fa2b09889b47ccd9a3e18