Difference between revisions of "Gamma function of x boxplus one"
From timescalewiki
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If $\mathbb{T}$ is a [[time scale]] and $s \in \mathbb{T}^+$, then for all $x \in \mathbb{R}^+$, | If $\mathbb{T}$ is a [[time scale]] and $s \in \mathbb{T}^+$, then for all $x \in \mathbb{R}^+$, | ||
$$\Gamma_{\mathbb{T}}\left(x \boxplus_{\mu} 1;s\right) = \dfrac{x}{s} \Gamma_{\mathbb{T}}(x;s),$$ | $$\Gamma_{\mathbb{T}}\left(x \boxplus_{\mu} 1;s\right) = \dfrac{x}{s} \Gamma_{\mathbb{T}}(x;s),$$ | ||
| − | where $\Gamma_{\mathbb{T}}$ denotes the [[gamma function]]. | + | where $\boxplus_{\mu}$ denotes [[forward box plus]] and $\Gamma_{\mathbb{T}}$ denotes the [[gamma function]]. |
==Proof== | ==Proof== | ||
Latest revision as of 18:08, 15 January 2023
Theorem
If $\mathbb{T}$ is a time scale and $s \in \mathbb{T}^+$, then for all $x \in \mathbb{R}^+$, $$\Gamma_{\mathbb{T}}\left(x \boxplus_{\mu} 1;s\right) = \dfrac{x}{s} \Gamma_{\mathbb{T}}(x;s),$$ where $\boxplus_{\mu}$ denotes forward box plus and $\Gamma_{\mathbb{T}}$ denotes the gamma function.
