A framework of Rogers–Ramanujan identities and their arithmetic properties

Abstract

The two Rogers–Ramanujan $q$-series

$${\sum }_{n=0}^{\infty }\frac{q^{n(n+\sigma )}}{(1-q)\cdots (1-q^n)},$$ 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

$${\sum }_{\lambda ,{\lambda }_1\le m}{q}^{a\left|\lambda \right|}{P}_{2\lambda }(1,q,q^2,\dots ;q^n)=\left[\mathrm{infinite}\mathrm{product}\mathrm{modular}\mathrm{function}\right],$$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 ${\mathrm{A}}_{2n}^{\left(2\right)}$ that the relevant $q$-series quotients are integral units.