$\Delta y = e^{sy}$ or: How I Learned to Stop Worrying and Love the $\Gamma$-function

TL;DR

This paper explores a novel functional equation involving exponential growth, constructing a specialized Gamma function for holomorphic functions and analyzing solutions to a complex difference equation.

Contribution

It introduces a new Gamma function associated with holomorphic functions and investigates solutions to an unconventional complex difference equation.

Findings

Construction of a holomorphic Gamma function for a given $f(s,z)$

Analysis of solutions to the difference equation $ riangle y = e^{sy}$ in the complex plane

Insights into functional equations and infinite compositions

Abstract

For a nice holomorphic function $f(s, z)$ in two variables, a respective holomorphic Gamma function $\Gamma = \Gamma_f$ is constructed, such that $f(s, \Gamma(s)) = \Gamma(s + 1)$. Along the way, we fall through a rabbit hole of infinite compositions, First Order Difference Equations, and absurd functional equations... This paper is orchestrated around an investigation into the unconventional equation $\Delta y = y(s+1)-y(s) = e^{sy(s)}$ and its solutions in the complex plane.