Vi\`ete's fractal distributions and their momenta

Abstract

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.