Is there a way I can prove this? This is part of one of my discrete math classes. I need to prove:
${\displaystyle \prod_{i\geq1} \frac{1}{1-xq^i}} = {\displaystyle \sum_{k\geq0} \frac{x^kq^{k^2}}{(1-q)(1-q^2)\cdots(1-q^k)(1-xq)(1-xq^2)\cdots(1-xq^k)}}$
One of the hints that was given was Let $O$ be the collection of integer partitions with only parts of odd size. Then both sides are the generating function $\sum_{p\epsilon O}x^{l(p)}q^{|p|}$ where $l(p)$ is the number of parts of $p$ and $|p|$ is the sum of the parts.
Im not sure if this hint is helpful but anything will be appreciated. The more detail the better. Thanks in advance!
Here is a visual proof of the slightly easier case when $x=q^a$.
The LHS counts all partitions with minimum part $a+1$.
The numerator on the RHS can be considered to be a 'Durfee rectangle' with sides $k$ and $k+a$. Place the first set of partitions on the $k$ side, and the second set aligned to the $k+a$ side, and again we have partitions into smallest part $a+1$.