Jacobi's triple product identity states that:
$\displaystyle \sum_{n = -\infty}^{\infty}z^{n}q^{n^{2}} = \prod_{n = 1}^{\infty}(1 - q^{2n})(1 + zq^{2n - 1})(1 + z^{-1}q^{2n - 1})$
I've seen a messy proof of this, but I still don't have any feeling for why it should be true. Is there some intuition behind it? (Or at least some reasonably nice combinatorial argument...)
The lecture notes of Igor Pak "Partition bijections, a survey" give several nice combinatorial proofs in chapter $6$ - Jacobi’s triple product identity: http://www.math.ucla.edu/~pak/papers/psurvey.pdf. It is perhaps a matter of taste what the intuition behind it is. This involves certainly much more than just a nice combinatorial argument, e.g., elliptic functions, Jacobi theta functions, etc., see the interesting discussion here: Motivation for/history of Jacobi's triple product identity.
