Entry 100

Level 8. Given \(q = e^{2\pi i \tau}\),  the q-Pochhammer symbol, and Ramanujan's functions \(f(a,b)\) and \(\psi(q)\) discussed in Entry 92. We have the Gollnitz-Gordon sum-product identities,

$$\begin{align}A(q) &=\frac{f(-q,-q^7)}{\psi(-q)} \,=\, \sum_{n=0}^\infty \frac {q^{n^2+2n}\,(-q;q^2)_n} {(q^2;q^2)_n} = \prod_{n=1}^\infty\frac{1}{(1-q^{8n-3})(1-q^{8n-4})(1-q^{8n-5})}\\ B(q) &= \frac{f(-q^3,-q^5)}{\psi(-q)} = \sum_{n=0}^\infty \frac {q^{n^2}\,(-q;q^2)_n} {(q^2;q^2)_n} \;\; = \;\; \prod_{n=1}^\infty\frac{1}{(1-q^{8n-1})(1-q^{8n-4})(1-q^{8n-7})}\end{align}$$

Let \(a=q^{7/16}A(q)\) and \(b= q^{-1/16}\,B(q)\). Then \((a, b)\), for appropriate \(\tau\), are actually radicals.