Let $\eta$ be the Dedekind eta function. Show that $\dfrac{\eta(q^9)^3}{\eta(q^3)}=\displaystyle\sum_{a,b\in \mathbb{Z}^2}q^{3(a^2+b^2+ab+a+b)+1}$.
I'm pretty sure the RHS is equal to $\theta_2(q^3)\psi_6(q^9)+\theta_3(q^3)\psi_3(q^9)$, but I'm not sure how to show this is equal to the LHS.
The equation is precisely equivalent to the cubic theta function identity (equation 2.1)
$$c(q^3) = \frac{(a(q)-b(q))}{3}$$
The proof appears on page 3 of "SOME CUBIC MODULAR IDENTITIES OF RAMANUJAN", J. M. Borwein, P. B. Borwein and F. G. Garvan, Trans. Amer. Math. Soc. 343 (1994), 35-47.