Vi\`ete's fractal distributions and their momenta
Abstract
Solutions of Schr\"oder-Poincar\'e's polynomial equations 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 and a set of discrete functions stable at infinity. The corresponding Vi\`ete-type infinite products for zeros of are also provided. This allows us to obtain a special kind of closed-form representation for 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 are computed explicitly.
Keywords
Cite
@article{arxiv.1906.04579,
title = {Vi\`ete's fractal distributions and their momenta},
author = {A. A. Kutsenko},
journal= {arXiv preprint arXiv:1906.04579},
year = {2020}
}