In a stage of proving the Jacobi triple product identity, I have to prove this :
For $|z|<1$ and $|q|<1$, we set $F_0(z,q)=1$ and $F_m(z,q)=\prod_{k=1}^m \frac{1}{1-q^kz}$.
If $(E_m)_{m\in\mathbb N}$ is a sequence of functions such that $\sum_{m=0}^\infty E_m(q)$ is absolutely convergent for any $|q|<1$, and if $$(\forall(z,q)\in\mathbb C^2)\ (|z|<1 \text{ and }|q|<1)\Longrightarrow \sum_{m=0}^\infty E_m(q)z^mF_m(z)=0$$then $E_m(q)=0$ for all $m\in\mathbb N$ and all $|q|<1$.
All I have is the relation $$F_m(zq,q)=(1-zq)\left[F_m(z,q)+zq^{m+1}F_{m+1}(z,q)\right]$$ and I don't even know how to use it (if it can be used). And the basic tool I'm allowed to use is the fact that if a power series cancels on a vicinity of $0$, then it's the null power series.
Can anyone point me in the right direction ?
Thanks in advance.