Identity involving q-Pochhammer symbols -- "Normalization"

Question

I am trying to prove the following identity involving q-Pochhammer symbols $$ \sum_{m=0}^{n} \dfrac{1}{(c^{-1};c^{-1})_m (c;c)_{n-m}}=1$$ where $n\in \mathbb{N}$, $c\in \mathbb{R}$ and $(a;q)_n=\prod_{i=0}^{n-1} (1-aq^i) $ denotes the q-Pochhammer symbol. I tried to run a few simulations for different values of $n$ and $c$ and I always got $1$, but I did not manage to prove it by using the induction principle and other identities. I think that the proof of this relation could give some insight to compute the other following summation $$ \sum_{m=0}^{n} \dfrac{c^{-mk}}{(c^{-1};c^{-1})_m (c;c)_{n-m}}$$ where $k\in \mathbb{N}$. Many thanks in advance to anyone having some suggestions.

Answer 1

Rearrange the summands and apply the q-binomial theorem: \begin{align*} \sum_{m=0}^n \frac{1}{(q^{-1};q^{-1})_m (q,q)_{n-m}} &= \sum_{m=0}^n \frac{(-1)^mq^{\frac{m(m+1)}{2}}}{(q;q)_m(q;q)_{n-m}} =\frac{(q;q)_n}{(q;q)_n}=1. \end{align*}