Difference between revisions of "Gamma function"

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We define  
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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
$$p_f(t,s)=e_{\frac{f}{\mathrm{id}}}(t,s),$$
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$$\Gamma_{\mathbb{T}}(f;s)=\mathscr{L}_{\mathbb{T}}\{p_{f \boxminus_{\mu} 1}(\cdot,s)\}(1)=\displaystyle\int_0^{\infty} p_{f \boxminus_{\mu}1}(\eta,s) e_{\ominus_{\mu}1}^{\sigma}(\eta,0) \Delta \eta,$$
where $\mathrm{id}$ denotes the identity map $\mathrm{id} \colon \mathbb{T} \rightarrow \mathbb{T}$ and $e_{\frac{f}{\mathrm{id}}}$ denotes the [[Delta exponential]]. Define the operations
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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]].
$$f \boxplus_{\mu} g := f+g+\dfrac{1}{\mathrm{id}}fg\mu$$
 
and
 
$$f \boxminus_{\mu} g := \dfrac{(f-g)\mathrm{id}}{\mathrm{id} + g \mu}.$$
 
With these definitions, we have the gamma operator <sup>[[#tgfots|[pp.516]]]</sup>
 
$$\Gamma_{\mathbb{T}}(f;s)=\mathscr{L}_{\mathbb{T}}\{p_{f \boxminus_{\mu} 1}(\cdot,s)\}(1)=\displaystyle\int_0^{\infty} p_{f \boxminus_{\mu}1}(\eta,s) e_{\ominus_{\mu}1}^{\sigma}(\eta,0) \Delta \eta.$$
 
  
 
=Properties of gamma functions=
 
=Properties of gamma functions=
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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[[Convergence of gamma function at positive values]]<br />
<strong>Theorem:</strong> If $s \in \mathbb{T}^+$, then $\Gamma_{\mathbb{T}}(x;s)$ converges for any $x \in \mathbb{R}^+$.
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[[Gamma function diverges at zero]]<br />
<div class="mw-collapsible-content">
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[[Gamma function diverges at infinity]]<br />
<strong>Proof:</strong>
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[[Gamma function equals one at one]]<br />
</div>
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[[Gamma function of x boxplus one]]<br />
</div>
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[[Gamma function on certain time scales at bracket number equals bracket factorial]]<br />
 
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
<strong>Theorem:</strong> If $s \in \mathbb{T}^+$, then
 
$$\displaystyle\lim_{x \rightarrow 0^+} \Gamma_{\mathbb{T}}(x;s) = \infty.$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
<strong>Theorem:</strong> If $s \in \mathbb{T}^+$, then
 
$$\displaystyle\lim_{x \rightarrow \infty} \Gamma_{\mathbb{T}}(x;s) = \infty.$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
<strong>Theorem:</strong> If $s \in \mathbb{T}^+$, then $\Gamma_{\mathbb{T}}(1;s)=1$.
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
<strong>Theorem:</strong> If $s \in \mathbb{T}^+$, then for all $x \in \mathbb{R}^+$,
 
$$\Gamma_{\mathbb{T}}(x \boxplus_{\mu} 1;s) = \dfrac{x}{s} \Gamma_{\mathbb{T}}(x;s).$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
Define the bracket number operators (they are actually functions)
 
$$[n]_{\mathbb{T}} = \left\{ \begin{array}{ll}
 
0 &; n=0 \\
 
[n-1]_{\mathbb{T}} \boxplus_{\mu} 1 &; n=1,2,\ldots
 
\end{array} \right.$$
 
and the bracket factorial
 
$$[n]_{\mathbb{T}}! = \left\{ \begin{array}{ll}
 
1&; n=0 \\
 
\displaystyle\prod_{j=1}^n [j]_{\mathbb{T}} &; n=1,2,\ldots
 
\end{array} \right.$$
 
 
 
<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 constant function on $\mathbb{T}^+$ for all $k\in[1,n]\bigcap \mathbb{Z}^+$. Then
 
$$\Gamma_{\mathbb{T}}\left( [n]_{\mathbb{T}};s \right) = \dfrac{[n-1]_{\mathbb{T}}!}{s^{n-1}}.$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
  
 
=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 $s \in \mathbb{T}^+$.
 +
<center>
 
{| class="wikitable"
 
{| 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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|$\dfrac{(q-1)s}{(1+(q-1)x)^{\log_q(s)}} \displaystyle\sum_{k=-\infty}^{\infty} \dfrac{(1+(q-1)x)^k}{\prod_{j=-\infty}^{k} (1+(q-1)q^k)}$
 
|$\dfrac{(q-1)s}{(1+(q-1)x)^{\log_q(s)}} \displaystyle\sum_{k=-\infty}^{\infty} \dfrac{(1+(q-1)x)^k}{\prod_{j=-\infty}^{k} (1+(q-1)q^k)}$
 
|}
 
|}
 +
</center>
  
 
=References=
 
=References=
<div id="tgfots"></div><bibtex>
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* {{PaperReference|The gamma function on time scales|2013|Martin Bohner|author2=Başak Karpuz|prev=Laplace transform|next=}}: Definition 2
@inproceedings{
+
 
  title="The Gamma Function on Time Scales",
+
[[Category:Definition]]
  author="Bohner, Martin and Karpuz, Başak",
 
  booktitle="Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis",
 
  volume="20",
 
  year="2013",
 
  pages="pp.507--522",
 
  url="http://online.watsci.org/abstract_pdf/2013v20/v20n4a-pdf/7.pdf"
 
}
 
</bibtex>
 

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