  Solutions of Schr\"oder-Poincar\'e's polynomial equations $f(az)=P(f(z))$
usually do not admit a simple closed-form representation in terms of known
standard functions. We show that there is a one-to-one correspondence between
zeros of $f$ and a set of discrete functions stable at infinity. The
corresponding Vi\`ete-type infinite products for zeros of $f$ are also
provided. This allows us to obtain a special kind of closed-form representation
for $f$ based on the Weierstrass-Hadamard factorization. From this
representation, it is possible to derive explicit momenta formulas for zeros.
We discuss also the rate of convergence of WH-factorization and momenta
formulas. Obtaining explicit closed-form expressions is the main motivation for
this work. Finally, all the branches of the multi-valued function $f^{-1}$ are
computed explicitly.
