An entire function connected with the approximation of the golden ratio

Abstract

In 1987, R. B. Paris uses the analytic function main g(w)=_n→∞(2φ)ⁿ(√1+√1+...√1+w_n-φ),\ \ \ φ=1+√52, to estimate the convergence of nested squares to the golden ratio. The function $g$ is non-entire and, perhaps, can not be expressed in terms of some standard known functions. We show that $f(z):=g^{-1}(z)$ is an entire function satisfying Poincare equality. While $f$