Today I gave my talk in the Proseminar on Partitions (Number Theory) offered by Prof. Kathrin Bringmann (University of Cologne). In my talk I gave a proof of the Jacobi Triple Product Identity and also proved some beautiful corollaries, e.g. Euler's Pentagonal Number Theorem.

Using the following definition of the Pochhammer Symbol

$(a;q)_n := \prod_{j=0}^{n-1}(1-aq^j)$

the Jacobi Triple Product Identity becomes

$\sum_{n\in \mathbb{Z}}z^nq^{n^2}=(q^2,q^2)_{\infty}(-zq,q^2)_{\infty}(-z^{-1}q,q^2)_{\infty}$ for $z\neq 0$

Using this equation, the Pentagonal Number Theorem

$(q;q)_{\infty}=\sum_{n\in \mathbb{Z}}(-1)^nq^{n(3n-1)/2}$

can be obtained by replacing $q$ by $q^{3/2}$.