Gamma function

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] $$\Gamma_{\mathbb{T}}(f;s)=\mathscr{L}_{\mathbb{T}}\{p_{f \boxminus_{\mu} 1}(\cdot,s)\}(1)=\displaystyle\int_0^{\infty} p_{f \boxminus_{\mu}1}(\eta,s) e_{\ominus_{\mu}1}^{\sigma}(\eta,0) \Delta \eta.$$

Examples of gamma functions

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

$\mathbb{T}=$	$\Gamma_{\mathbb{T}}(x;s)=$
$\mathbb{R}_0^+$	$\displaystyle\int_0^{\infty} \left( \dfrac{\tau}{s} \right)^{x-1}e^{-\tau} d\tau$
$h\mathbb{N}_0;h>0$	$h \displaystyle\sum_{k=0}^{\infty} \left( \displaystyle\prod_{j=s}^{k-1} \dfrac{j+x}{j+1} \right) \dfrac{1}{(1+h)^{k+1}}$
$\overline{q^{\mathbb{Z}}}; q>1$	$\dfrac{(q-1)s}{(1+(q-1)x)^{\log_q(s)}} \displaystyle\sum_{k=-\infty}^{\infty} \dfrac{(1+(q-1)x)^k}{\prod_{j=-\infty}^{k} (1+(q-1)q^k)}$

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>