Entry 92

Given the q-Pochhammer symbol, $$\begin{align}(a;q)_n &= \prod_{k=0}^{n-1}(1-aq^k)\\ (a;q)_\infty &= \prod_{k=0}^{\infty}(1-aq^k)\end{align}$$

as well as the Ramanujan theta function,

$$f(a,b) = \sum_{n=-\infty}^\infty a^{n(n+1)/2} \; b^{n(n-1)/2}$$

In his Notebooks, Ramanujan also defined four one-parameter versions he commonly used as,

$$\begin{align}\varphi(q) &= f(q,\, q)\\ f(-q) &= f(-q, -q^2)\\ \psi(q) &= f(q,\, q^3)\\ \chi(q) &=\frac{f(-q^2,-q^2)}{f(-q,-q^2)}\\ \end{align}$$

which we will also use in later entries. There are several simple relations between these four auxiliary functions, one of which I found using all four is the elegant Fermat curve of degree 8,

$$\big[f(-q)\,\chi(-q)\big]^8+\big[\sqrt2\,q^{1/8}\psi(-q)\big]^8 = \big[\varphi(-q^2)\big]^8$$

We normally assume \(q = e^{2\pi i \tau}\) unless otherwise specified. For simplicity, it will also be assumed \(\tau\) is an complex quadratic number so that certain functions later will also evaluate as radicals.