A framework of Rogers–Ramanujan identities and their arithmetic properties

Abstract

Abstract. The two Rogers-Ramanujan q-serieswhere 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 doublyinfinite families of q-series identities. If a ∈ {1, 2} and m, n ≥ 1, then we have λ λ1≤mwhere the P λ (x 1 , x 2 , . . . ; 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 that the relevant q-series quotients are integral units.

Combinatorial Properties of Rogers-Ramanujan Type Identities Arising from Hall-Littlewood Polynomials

“…First, we show that this framework gives rise to a recursive method of computing the coefficients of these q-series, which allows us to directly compute these q-series. [2] proves that the expression in (1.3) equals an infinite product of terms (1 − q n ). For the specific pairs ν in {(1, −1), (2, −1), (1, 0), (2, −2)}, the pairs considered in [2], we define c ν (a, b; t) as follows:…”

“…[2] proves that the expression in (1.3) equals an infinite product of terms (1 − q n ). For the specific pairs ν in {(1, −1), (2, −1), (1, 0), (2, −2)}, the pairs considered in [2], we define c ν (a, b; t) as follows:…”

“…The work of Griffin, Ono, and Warnaar shows that for special choices of ν, R ν (a, b, q) equals an infinite product of modular functions [2]. Notice that this framework then gives a way to define the q-series sides of the Rogers-Ramanujan type identities in terms of symmetric functions.…”

“…The famous Rogers-Ramanujan identities [3, p. 384] have inspired discoveries in many fields of mathematics and physics, such as algebraic geometry, group theory, knot theory, modular forms, orthogonal polynomials, statistical mechanics, probability, and transcendental number theory (see references in [2]). Much of their importance stems from the fact that the Rogers-Ramanujan infinite sums correspond to q-series which happen to equal infinite products.…”

“…Recently, Griffin, Ono, and Warnaar unearthed a more general framework for Rogers-Ramanujan-type identities where the infinite sum of symmetric functions is equal to an infinite product with periodic exponents, which turns out essentially to be a modular function. These series are defined using Hall-Littlewood polynomials P λ (x; q) (see Section 3.1 for definitions) [2].…”