Difference between revisions of "Gamma function"

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(Examples of gamma functions)
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=Properties of gamma functions=
 
=Properties of gamma functions=
 +
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 +
<strong>Theorem:</strong> If $s \in \mathbb{T}^+$, then $\Gamma_{\mathbb{T}}(x;s)$ converges for any $x \in \mathbb{R}^+$.
 +
<div class="mw-collapsible-content">
 +
<strong>Proof:</strong> █
 +
</div>
 +
</div>
  
==Convergence==
+
<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=
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|$\Gamma_{\mathbb{T}}(x;s)=$
 
|$\Gamma_{\mathbb{T}}(x;s)=$
 
|-
 
|-
|$\mathbb{R}_0^+$
+
|[[Real_numbers | $\mathbb{R}$]]
 
|$\displaystyle\int_0^{\infty} \left( \dfrac{\tau}{s} \right)^{x-1}e^{-\tau} d\tau$
 
|$\displaystyle\int_0^{\infty} \left( \dfrac{\tau}{s} \right)^{x-1}e^{-\tau} d\tau$
 
|-
 
|-
|$h\mathbb{N}_0;h>0$
+
|[[Multiples_of_integers | $h\mathbb{Z};h>0$]]
 
|$h \displaystyle\sum_{k=0}^{\infty} \left( \displaystyle\prod_{j=s}^{k-1} \dfrac{j+x}{j+1} \right) \dfrac{1}{(1+h)^{k+1}}$
 
|$h \displaystyle\sum_{k=0}^{\infty} \left( \displaystyle\prod_{j=s}^{k-1} \dfrac{j+x}{j+1} \right) \dfrac{1}{(1+h)^{k+1}}$
 
|-
 
|-
|$\overline{q^{\mathbb{Z}}}; q>1$
+
|[[Quantum_q_greater_than_1_integerpower | $\overline{q^{\mathbb{Z}}}; q>1$]]
 
|$\dfrac{(q-1)s}{(1+(q-1)x)^{\log_q(s)}} \displaystyle\sum_{k=-\infty}^{\infty} \dfrac{(1+(q-1)x)^k}{\prod_{j=-\infty}^{k} (1+(q-1)q^k)}$
 
|$\dfrac{(q-1)s}{(1+(q-1)x)^{\log_q(s)}} \displaystyle\sum_{k=-\infty}^{\infty} \dfrac{(1+(q-1)x)^k}{\prod_{j=-\infty}^{k} (1+(q-1)q^k)}$
 
|}
 
|}

Revision as of 03:06, 24 July 2014

We 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 $e_{\cdot}$ denotes the time scale exponential. Define the operations $$f \boxplus_{\mu} g := f+g+\dfrac{1}{\mathrm{id}}fg\mu$$ and $$f \boxminus_{\mu} g := \dfrac{(f-g)\mathrm{id}}{\mathrm{id} + g \mu}.$$ With these definitions, we have the gamma operator [pp.516] $$\Gamma_{\mathbb{T}}(f;s)=\mathscr{L}_{\mathbb{T}}\{p_{f \boxminus_{\mu} 1}(\cdot,s)\}(1)=\displaystyle\int_0^{\infty} p_{f \boxminus_{\mu}1}(\eta,s) e_{\ominus_{\mu}1}^{\sigma}(\eta,0) \Delta \eta.$$

Properties of gamma functions

Theorem: If $s \in \mathbb{T}^+$, then $\Gamma_{\mathbb{T}}(x;s)$ converges for any $x \in \mathbb{R}^+$.

Proof:

Theorem: If $s \in \mathbb{T}^+$, then $$\displaystyle\lim_{x \rightarrow 0^+} \Gamma_{\mathbb{T}}(x;s) = \infty.$$

Proof:

Theorem: If $s \in \mathbb{T}^+$, then $$\displaystyle\lim_{x \rightarrow \infty} \Gamma_{\mathbb{T}}(x;s) = \infty.$$

Proof:

Theorem: If $s \in \mathbb{T}^+$, then $\Gamma_{\mathbb{T}}(1;s)=1$.

Proof:

Theorem: If $s \in \mathbb{T}^+$, then for all $x \in \mathbb{R}^+$, $$\Gamma_{\mathbb{T}}(x \boxplus_{\mu} 1;s) = \dfrac{x}{s} \Gamma_{\mathbb{T}}(x;s).$$

Proof:

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 $$[n]_{\mathbb{T}}! = \left\{ \begin{array}{ll} 1&; n=0 \\ \displaystyle\prod_{j=1}^n [j]_{\mathbb{T}} &; n=1,2,\ldots \end{array} \right.$$

Theorem: 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 $$\Gamma_{\mathbb{T}}\left( [n]_{\mathbb{T}};s \right) = \dfrac{[n-1]_{\mathbb{T}}!}{s^{n-1}}.$$

Proof:

Examples of gamma functions

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

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

References

<bibtex>

@inproceedings{

  title="The Gamma Function on Time Scales",
  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>