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I have done a fair bit of thinking on this. Lets say that $z(f(x))=f(x+1)-f(x)$. I figured out that $z\left(\frac{x^2-x}{2}\right)=x$. But one over that does not equal $\frac{1}{x}$.

Robert Lee
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ggbg
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    dear ggbg, I have seen you at mathoverflow asking about tetration.
    1. going to mathoverflow when your reputation or experience is very low is something i do not recommend.
    2. as for tetration , you asked for a plot , but there are infinitely many tetrations !! So as stated the question was not solvable.
    3. For tetration there is the tetration forum
     – mick Feb 18 '23 at 22:48

1 Answers1

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Yes. Let $L(z)$ denote the logarithm of the gamma function.

Then $L(x + 1) - L(x) = \log(x)$. So the derivative $L'(x)$ (i.e. $\Gamma'(x) / \Gamma(x)$) satisfies the desired functional equation.

hunter
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