Difference between revisions of "Gamma function on certain time scales at bracket number equals bracket factorial"

Revision as of 18:06, 15 January 2023

Theorem

If $\mathbb{T}$ is a time scale, $n \in \mathbb{Z}^+$, and $[k]_{\mathbb{T}}$ is constant on $\mathbb{T}^+$ for all $k\in[1,n]\bigcap \mathbb{Z}^+$, then $$\Gamma_{\mathbb{T}}\left( [n]_{\mathbb{T}};s \right) = \dfrac{[n-1]_{\mathbb{T}}!}{s^{n-1}},$$ where $\Gamma$ denotes the gamma function.

Proof

References