Difference between revisions of "Gamma function on certain time scales at bracket number equals bracket factorial"
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(Created page with "==Theorem== <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> <strong>Theorem:</strong> Let $n \in \mathbb{Z}^+$ and assume that $[k]_{\mathbb{T}}$ is a...") |
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==Theorem== | ==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 | |
− | + | $$\Gamma_{\mathbb{T}}\left( [n]_{\mathbb{T}};s \right) = \dfrac{[n-1]_{\mathbb{T}}!}{s^{n-1}},$$ | |
− | $$\Gamma_{\mathbb{T}}\left( [n]_{\mathbb{T}};s \right) = \dfrac{[n-1]_{\mathbb{T}}!}{s^{n-1}} | + | where $\Gamma$ denotes the [[gamma function]]. |
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==Proof== | ==Proof== |
Revision as of 18:06, 15 January 2023
Theorem
If $\mathbb{T}$ is a time scale, $n \in \mathbb{Z}^+$, and $[k]_{\mathbb{T}}$ is constant on $\mathbb{T}^+$ for all $k\in[1,n]\bigcap \mathbb{Z}^+$, then $$\Gamma_{\mathbb{T}}\left( [n]_{\mathbb{T}};s \right) = \dfrac{[n-1]_{\mathbb{T}}!}{s^{n-1}},$$ where $\Gamma$ denotes the gamma function.