Affine Algebra Unifies Rogers-Ramanujan and Andrews–Gordon Identities — E8 Intelligence Research

FINDING: The A_{2n}^{(2)} Rogers-Ramanujan identities form a doubly-infinite family where product sides are specialised characters of affine Kac-Moody algebra A_{2n}^{(2)} at level m, unifying classical Rogers-Ramanujan and Andrews–Gordon identities. | MATH: For positive integers m, n, the product side is χ_{A_{2n}^{(2)}}(level m, specialised); the identities equate this to a sum side of q-series with exponents governed by the affine root system. The classical Rogers-Ramanujan is the n=1, m=2 case (product ∏_{k≥1, k≠0,±1 mod 5} 1/(1−q^k)). The Andrews–Gordon corresponds to specific (m,n) pairs. The A_{2n}^{(2)} root system has Dynkin diagram with n nodes, one long root, and the Weyl group is the affine Weyl group of type C_n^{(1)} — its Cartan matrix has determinant 2n+1, and the level m character involves the modular form η(τ)^{2n+1} in the denominator. | CONNECTION: The determinant 2n+1 directly yields the modulus 2n+1 in the product side (e.g., 5 for n=1, 7 for n=2, 9 for n=3). This Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com