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For a nice holomorphic function $f(s, z)$ in two variables, a respectiveholomorphic 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 ofinfinite compositions, First Order Difference Equations, and absurd functionalequations... This paper is orchestrated around an investigation into theunconventional equation $\Delta y = y(s+1)-y(s) = e^{sy(s)}$ and its solutionsin the complex plane.

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