Gamma function of x boxplus one

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Theorem

If $\mathbb{T}$ is a time scale and $s \in \mathbb{T}^+$, then for all $x \in \mathbb{R}^+$, $$\Gamma_{\mathbb{T}}(x \boxplus_{\mu} 1;s) = \dfrac{x}{s} \Gamma_{\mathbb{T}}(x;s),$$ where $\Gamma_{\mathbb{T}}$ denotes the gamma function.

Proof

References