JACOBI TRIPLE PRODUCT AND PRODUCT REPRESENTATION OF JACOBI THETA FUNCTIONS

In this entry I will prove the remarkable Jacobi triple product using one of the most elementary proofs due to great mathematician George Andrews.

\begin{align*} \sum_{n=-\infty}^{\infty} q^{n^{2}} z^{n}=\prod_{n=1}^{\infty}\left(1+z q^{2 n-1}\right)\left(1+z^{-1} q^{2 n-1}\right)\left(1-q^{2 n}\right) \end{align*}

Additionally, it will be shown some product representations of Jacobi theta function and the beautiful Euler´s pentagonal theorem

\begin{align*} \prod_{n=1}^\infty(1-q^n)=\sum_{n=-\infty}^{\infty}(-1)^nq^{(3n+1)n/2} \end{align*}

This note is an excerpt of my ongoing personal project of putting all the blog content with additional content in a book format, so if you like it or would like to leave a comment please let me know. Click here to the proof.

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