$Δy = e^{sy}$ or: How I Learned to Stop Worrying and Love the $Γ$-function

James David Nixon

paper

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

arXiv:1910.05111

Abstract

For a nice holomorphic function $f (s, z)$ in two variables, a respective holomorphic Gamma function $Γ = Γ_{f}$ is constructed, such that $f (s, Γ (s)) = Γ (s + 1)$ . 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 $Δ y = y (s + 1) - y (s) = e^{sy (s)}$ and its solutions in the complex plane.

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Δy=esy or: How I Learned to Stop Worrying and Love the Γ-function

Abstract

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