Ramanujan Identity related to JacobiFunction

Question

The following identity is allegedly due to Ramanujan $$\int_0^\infty \frac{{\rm d}x}{(1+x^2)(1+r^2x^2)(1+r^4x^2)\cdots} = \frac{\pi/2}{\sum_{n=0}^\infty r^{\frac{n(n+1)}{2}}} \, $$but how do you prove this? The denominator of the right side is related to the Jacobi Function, so maybe one could proceed via modular forms?

Answer 1

A partial answer for now. We have to prove that $$ \prod_{n\geq 1}\frac{1}{1+r^n}=\sum_{k\geq 0}\prod_{n=1}^{k}\frac{r^{2n-1}}{r^{2n}-1} $$ or $$ \prod_{n\geq 1}\frac{1-r^n}{1-r^{2n}}=\sum_{k\geq 0}(-1)^k r^{k^2} \prod_{n=1}^{k}\frac{1}{1-r^{2n}} $$ or $$ \prod_{n\geq 1}(1-r^n) = \sum_{k\geq 0}(-1)^k r^{k^2} \prod_{n>k}(1-r^{2n}) $$

where the LHS, by Euler's pentagonal number theorem, equals $$\sum_{k=-\infty}^{+\infty}(-1)^k r^{k(3k-1)/2} $$ and the coefficient of $r^m$ in $\prod_{n>k}(1-r^n)$ depends on the number of partitions of $m$ into distinct parts with cardinality $>k$, accounted with a positive or negative sign according to the number of parts.

Now it shouldn't be difficult to prove our claim by using the same involution exploited in the combinatorial proof of Euler's pentagonal number theorem, or something quite close to it.