Difference between revisions of "Gamma function on certain time scales at bracket number equals bracket factorial"
From timescalewiki
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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 | 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 | ||
$$\Gamma_{\mathbb{T}}\left( [n]_{\mathbb{T}};s \right) = \dfrac{[n-1]_{\mathbb{T}}!}{s^{n-1}},$$ | $$\Gamma_{\mathbb{T}}\left( [n]_{\mathbb{T}};s \right) = \dfrac{[n-1]_{\mathbb{T}}!}{s^{n-1}},$$ | ||
| − | where $\Gamma$ denotes the [[gamma function]]. | + | where $[k]_{\mathbb{T}}$ denotes a [[bracket number]], $[n-1]_{\mathbb{T}}!$ denotes a [[bracket factorial]], and $\Gamma$ denotes the [[gamma function]]. |
==Proof== | ==Proof== | ||
Latest revision as of 18:07, 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 $[k]_{\mathbb{T}}$ denotes a bracket number, $[n-1]_{\mathbb{T}}!$ denotes a bracket factorial, and $\Gamma$ denotes the gamma function.
