Andrews-Baxter Proof of Rogers-Ramanujan Identities and Ramanujan's Partition Congruences — E8 Intelligence Research

Abstract

FINDING: Rogers-Ramanujan identities proven via Andrews-Baxter motivated proof; Ramanujan's unpublished manuscript on partition and tau function congruences. MATH: Rogers-Ramanujan identities: \[ \prod_{k=0}^{\infty} \frac{1}{(1 - q^{5k+1})(1 - q^{5k+4})} = \sum_{n=0}^{\infty} \frac{q^{n^2}}{(1-q)(1-q^2)\cdots(1-q^n)} \] \[ \prod_{k=0}^{\infty} \frac{1}{(1 - q^{5k+2})(1 - q^{5k+3})} = \sum_{n=0}^{\infty} \frac{q^{n^2+n}}{(1-q)(1-q^2)\cdots(1-q^n)} \] Ramanujan's partition congruences: \( p(5n+4) \equiv 0 \mod 5 \), \( p(7n+5) \equiv 0 \mod 7 \), \( p(11n+6) \equiv 0 \mod 11 \). Tau function congruences: \( \tau(n) \) modulo small primes (e.g., \( \tau(n) \equiv n\sigma_3(n) \mod 5 \)). CONNECTION: The modulus 5 in Rogers-Ramanujan links to the golden ratio \(\phi = (1+\sqrt{5})/2 \approx 1.618\) and its reciprocal \(1/\phi \approx 0.618\), as the identities involve 5-fold symmetry. The partition congruences modulo 5,7,11 relate to the exceptional Lie algebra root systems \( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com