Questions tagged [q-series]

I have to prove the following identity: $$\sum_{n\geq 0} (-1)^n(2n+1)q^{\frac{n(n+1)}{2}} = (q;q)_\infty^3$$ where $(a;q)_\infty = \prod_{i\geq 0}1-aq^i$ is the $q$-Pochhammer symbol. In my notes the identity is stated as a corollary of Jacobi…

Here is my first question in this site Prove the following $$\lim_{q \to 1 }\frac{(a)_{\infty}}{(aq^x)_{\infty}}=(1-a)^x$$ If $x$ was an integer , then this is an easy task.

Prove the following $$\frac{1}{(z;q)_{\infty}} = \sum^{\infty}_{k=0} \frac{z^k}{(q;q)_k}$$ I am looking for a proof that doesn't involve the q-binomial theorem . where $$(a;q)_k = \prod_{n=0}^{k-1}(1-aq^n)$$ Any help is really appreciated.