Let $\beta,\omega\in(0,1)$ and $z>0$ then the following seems to be true: $$\sum_{j=0}^{\infty}{\beta^j \frac{\prod_{k=0}^{j}{\frac{1+z(\beta\omega)^{-k}}{1+z(\beta\omega)^{1-k}}}}{1+z(\beta\omega)^{-j}}}=\frac{1}{(1-\beta)(1+z\beta\omega)}$$
This identity pops up as something that intuitively ought to hold in an Econ problem, and I have verified it for particular $\beta,\omega,z$ in Maple.
I have no idea how to prove it though. Any suggestions (or a counter-example) would be much appreciated!
This is actually quite easy as one notices that all coefficients above are actually equal to $\frac{1}{1+zbw}$ by simplifying the numerators and denominators successively (they are of the type $\frac{1}{f(-j)}\Pi_{k=0}^j\frac{f(-k)}{f(-k+1)}=1/f(1)$) so the expression is $$ \frac{1}{1+zbw}\sum_{j=0}^{\infty}\beta^j=\frac{1}{(1-\beta)(1+z\beta\omega)}$$