Difference between revisions of "Gamma function of x boxplus one"

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==Theorem==
 
==Theorem==
 
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 $x \in \mathbb{R}^+$,
$$\Gamma_{\mathbb{T}}(x \boxplus_{\mu} 1;s) = \dfrac{x}{s} \Gamma_{\mathbb{T}}(x;s),$$
+
$$\Gamma_{\mathbb{T}}\left(x \boxplus_{\mu} 1;s\right) = \dfrac{x}{s} \Gamma_{\mathbb{T}}(x;s),$$
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.

Proof

References