Difference between revisions of "Gamma function"
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=Examples of gamma functions= | =Examples of gamma functions= | ||
| + | We write formulas for gamma functions defined for $x \in \mathbb{R}^+$ and $s \in \mathbb{T}^+$. | ||
| + | {| class="wikitable" | ||
| + | |- | ||
| + | |$\mathbb{T}=$ | ||
| + | |$\Gamma_{\mathbb{T}}(x;s)=$ | ||
| + | |- | ||
| + | |$\mathbb{R}_0^+$ | ||
| + | |$\displaystyle\int_0^{\infty} \left( \dfrac{\tau}{s} \right)^{x-1}e^{-\tau} d\tau$ | ||
| + | |- | ||
| + | |$h\mathbb{N}_0;h>0$ | ||
| + | |$h \displaystyle\sum_{k=0}^{\infty} \left( \displaystyle\prod_{j=s}^{k-1} \dfrac{j+x}{j+1} \right) \dfrac{1}{(1+h)^{k+1}}$ | ||
| + | |- | ||
| + | |$\overline{q^{\mathbb{Z}}}; q>1$ | ||
| + | |$\dfrac{(q-1)s}{(1+(q-1)x)^{\log_q(s)}} \displaystyle\sum_{k=-\infty}^{\infty} \dfrac{(1+(q-1)x)^k}{\prod_{j=-\infty}^{k} (1+(q-1)q^k)}$ | ||
| + | |} | ||
=References= | =References= | ||
Revision as of 02:53, 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
Convergence
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>
