A framework of Rogers-Ramanujan identities and their arithmetic properties

Abstract

The two Rogers-Ramanujan $q$-series \[ \sum_{n=0}^{\infty}\frac{q^{n(n+σ)}}{(1-q)\cdots (1-q^n)}, \] where $\sigma =0,1$, 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 $a\in \{1,2\}$ and $m,n\ge 1$, then we have \[ \sum_{\substack{λλ_1\leq m}} q^{a|λ|} P_{2λ}(1,q,q^2,\dots;q^n) =\textrm{"infinite product modular function"}, \] where the $P_{\lambda }(x_1,x_2,\dots ;q)$ 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 ${\text{A}}_{2n}^{(2)}$ that the relevant $q$-series quotients are integral units.

44 pages. This paper supersedes the paper arXiv:1309.5216 "The A_{2n}^{(2)} Rogers-Ramanujan identities"; Improved introduction, typos corrected and references added; Final version to appear in Duke Mathematical Journal; Typographical errors are corrected

References in corpus (3)

• Rogers-Ramanujan type identities and the head and tail of the colored Jones polynomial
• q-series and L-functions related to half-derivatives of the Andrews--Gordon identity
• Quasimodular forms and sl(m|m)^ characters

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• Topological strings, quiver varieties and Rogers-Ramanujan identities
• A motivated proof of the Göllnitz-Gordon-Andrews identities
• Generalisations of Capparelli's and Primc's identities, II: perfect $A_{n-1}^{(1)}$ crystals and explicit character formulas
• Schur partition theorems via perfect crystal
• A family of vector-valued quantum modular forms of depth two
• Generalized Andrews-Gordon Identities
• Partition identities from higher level crystals of $A_1^{(1)}$
• Lepowsky's and Wakimoto's product formulas for the affine Lie algebras $C_l^{(1)}$
• Modular units from quotients of Rogers-Ramanujan type $q$-series
• Combinatorial Properties of Rogers-Ramanujan-Type Identities Arising from Hall-Littlewood Polynomials