Difference between revisions of "Gamma function"

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(Properties of gamma functions)
 
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[[Gamma function equals one at one]]<br />
 
[[Gamma function equals one at one]]<br />
 
[[Gamma function of x boxplus one]]<br />
 
[[Gamma function of x boxplus one]]<br />
[[Gamma function on integers at bracket number equals bracket factorial]]<br />
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[[Gamma function on certain time scales at bracket number equals bracket factorial]]<br />
  
 
=Examples of gamma functions=
 
=Examples of gamma functions=
 
We write formulas for gamma functions defined for $x \in \mathbb{R}^+$ and $s \in \mathbb{T}^+$.
 
We write formulas for gamma functions defined for $x \in \mathbb{R}^+$ and $s \in \mathbb{T}^+$.
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{| class="wikitable"
 
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|[[Real_numbers | $\mathbb{R}$]]
 
|[[Real_numbers | $\mathbb{R}$]]
|$\displaystyle\int_0^{\infty} \left( \dfrac{\tau}{s} \right)^{x-1}e^{-\tau} d\tau$
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|$\displaystyle\int_0^{\infty} \left( \dfrac{\tau}{s} \right)^{x-1}e^{-\tau} \mathrm{d}\tau$
 
|-
 
|-
 
|[[Multiples_of_integers | $h\mathbb{Z};h>0$]]
 
|[[Multiples_of_integers | $h\mathbb{Z};h>0$]]
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|$\dfrac{(q-1)s}{(1+(q-1)x)^{\log_q(s)}} \displaystyle\sum_{k=-\infty}^{\infty} \dfrac{(1+(q-1)x)^k}{\prod_{j=-\infty}^{k} (1+(q-1)q^k)}$
 
|$\dfrac{(q-1)s}{(1+(q-1)x)^{\log_q(s)}} \displaystyle\sum_{k=-\infty}^{\infty} \dfrac{(1+(q-1)x)^k}{\prod_{j=-\infty}^{k} (1+(q-1)q^k)}$
 
|}
 
|}
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</center>
  
 
=References=
 
=References=
 
* {{PaperReference|The gamma function on time scales|2013|Martin Bohner|author2=Başak Karpuz|prev=Laplace transform|next=}}: Definition 2
 
* {{PaperReference|The gamma function on time scales|2013|Martin Bohner|author2=Başak Karpuz|prev=Laplace transform|next=}}: Definition 2
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[[Category:Definition]]

Latest revision as of 12:53, 16 January 2023

Let $\mathbb{T}$ be a time scale and define $p_f(t,s)=e_{\frac{f}{\mathrm{id}}}(t,s)$, where $\mathrm{id}$ denotes the identity map $\mathrm{id} \colon \mathbb{T} \rightarrow \mathbb{T}$ and $e_{\frac{f}{\mathrm{id}}}$ denotes the delta exponential. With these definitions, we define the gamma operator $$\Gamma_{\mathbb{T}}(f;s)=\mathscr{L}_{\mathbb{T}}\{p_{f \boxminus_{\mu} 1}(\cdot,s)\}(1)=\displaystyle\int_0^{\infty} p_{f \boxminus_{\mu}1}(\eta,s) e_{\ominus_{\mu}1}^{\sigma}(\eta,0) \Delta \eta,$$ where $\mathscr{L}_{\mathbb{T}}$ denotes the Laplace transform, $\boxminus_{\mu}$ denotes forward box minus, $\ominus_{\mu}$ denotes forward circle minus, and $\sigma$ denotes the forward jump.

Properties of gamma functions

Convergence of gamma function at positive values
Gamma function diverges at zero
Gamma function diverges at infinity
Gamma function equals one at one
Gamma function of x boxplus one
Gamma function on certain time scales at bracket number equals bracket factorial

Examples of gamma functions

We write formulas for gamma functions defined for $x \in \mathbb{R}^+$ and $s \in \mathbb{T}^+$.

$\mathbb{T}=$ $\Gamma_{\mathbb{T}}(x;s)=$
$\mathbb{R}$ $\displaystyle\int_0^{\infty} \left( \dfrac{\tau}{s} \right)^{x-1}e^{-\tau} \mathrm{d}\tau$
$h\mathbb{Z};h>0$ $h \displaystyle\sum_{k=0}^{\infty} \left( \displaystyle\prod_{j=s}^{k-1} \dfrac{j+x}{j+1} \right) \dfrac{1}{(1+h)^{k+1}}$
$\overline{q^{\mathbb{Z}}}; q>1$ $\dfrac{(q-1)s}{(1+(q-1)x)^{\log_q(s)}} \displaystyle\sum_{k=-\infty}^{\infty} \dfrac{(1+(q-1)x)^k}{\prod_{j=-\infty}^{k} (1+(q-1)q^k)}$

References