I have honestly tried a variety of things for this identity. I just straight plugged stuff in and tried to simplify as much as possible (for a while), and I also tried an induction proof, but I cannot prove this identity.
From Bruce Berndt's "Number Theory in the Spirit of Ramanujan."
Prove:
$$f(a,b)=a^{n(n+1)/2}b^{n(n-1)/2}f(a(ab)^n,b(ab)^{-n})$$
Where $n$ is any positive integer, and,
$$f(a,b) = \sum_{j=-\infty}^{\infty}a^{j(j+1)/2}b^{j(j-1)/2}$$
ie Ramanujan's Theta Function.
I'm looking for a plan of attack more than anything.
It's just a matter of substitution and simplification:
$$ a^{n(n+1)/2} b^{n(n-1)/2} f(a {(ab)}^n, b {(ab)}^{-n}) = a^{n(n+1)/2} b^{n(n-1)/2} \sum_{j = -\infty}^{\infty} a^{j(j+1)/2 + jn}\; b^{j(j-1)/2 + j n} $$
Since the index $j$ goes from $-\infty$ to $+\infty$, we can shift it however we like. So: Let $j\rightarrow j-n$, and then simplify the exponents of $a$ and $b$. You'll find you get $f(a,b)$ back.