Entry 38

For consistency, let the variable \(q\) be the nome's square \(q=e^{2\pi i \tau}\,\)throughout. 

I. The null Jacobi theta functions (with \(z=0\)) are,
$$\quad \vartheta_3(q) = \sum_{m=-\infty}^{\infty}q^{n^2} = \frac{\eta^5(2\tau)}{\eta^2(\tau)\eta^2(4\tau)}$$$$
\vartheta_4(q) = \sum_{m=-\infty}^{\infty}(-1)^n\,q^{n^2} = \frac{\eta^2(\tau)}{\eta(2\tau)}$$$$
\vartheta_2(q) = \sum_{m=-\infty}^{\infty}q^{(n+1/2)^2} = \frac{2\,\eta^2(4\tau)}{\eta(2\tau)}$$
II. The Borwein cubic theta functions are,
$$a(q)=\sum_{m,n=-\infty}^{\infty}q^{m^2+mn+n^2} = \frac{\eta^3(\tau)}{\eta(3\tau)} +\frac{9\,\eta^3(9\tau)}{\eta(3\tau)}$$$$
b(q)=\sum_{m,n=-\infty}^{\infty}\zeta^{m-n}q^{m^2+mn+n^2} = \frac{\eta^3(\tau)}{\eta(3\tau)}\quad\quad$$$$
\quad c(q)=\sum_{m,n=-\infty}^{\infty}q^{(m+1/3)^2+(m+1/3)(n+1/3)+(n+1/3)^2} = \frac{3\,\eta^3(3\tau)}{\eta(\tau)}$$
with a cube root of unity \(\zeta =e^{2\pi i/3}\). 

III. The derived Jacobi theta functions are,
$$\qquad\qquad d(q)= \vartheta_4^4(q)+2\,\vartheta_2^4(q) = \frac{\eta^8(\tau)}{\eta^4(2\tau)} +\frac{32\,\eta^8(4\tau)}{\eta^4(2\tau)}$$$$
\quad e(q)= \vartheta_4^4(q)= \frac{\eta^8(\tau)}{\eta^4(2\tau)}$$$$
\quad f(q)= \tfrac12 \vartheta_2^4(q^{1/2})= \frac{8\,\eta^8(2\tau)}{\eta^4(\tau)}$$ These obey beautiful relations discussed in Entry 39.