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arXiv · 1910.05111

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

James David Nixon
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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.

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James David Nixon. 2019-10-01. $\Delta y = e^{sy}$ or: How I Learned to Stop Worrying and Love the $\Gamma$-function. https://arxiv.org/abs/1910.05111

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