q-Binomial Theorem and Rogers-Ramanujan Identities: A Scattered Search — E8 Intelligence Research

FINDING: The search results are a scattered mix of pedagogical videos on q-binomial identities and Rogers-Ramanujan identities, plus an unrelated QCD experimental report; no direct source on the hyperoctahedral group \(B_n\) Weyl denominator identity was retrieved. The mathematical core is the classical q-binomial theorem and its role in Rogers-Ramanujan-type sum-product identities. MATH: - q-binomial theorem: \(\sum_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix}_q q^{\binom{k}{2}} x^k = \prod_{i=0}^{n-1} (1 + x q^i)\), where \(\begin{bmatrix} n \\ k \end{bmatrix}_q = \frac{(q;q)_n}{(q;q)_k (q;q)_{n-k}}\). - Rogers-Ramanujan identities (first): \(\sum_{k=0}^\infty \frac{q^{k^2}}{(q;q)_k} = \frac{1}{(q;q^{5})_\infty (q^4;q^{5})_\infty}\). - Weyl denominator identity for \(B_n\) (implied, not in results): \(\prod_{\alpha \in \Phi^+} (1 - e^{-\alpha}) = \sum_{w \in W} (-1)^{\ell(w)} e^{w(\rho) - \rho}\), with \(\Phi^+\) positive roots of \(B_n\), \(W\) the hyperoctahedral group of or Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com