Difference between revisions of "Gamma function"
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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 | |
| − | + | $$\Gamma_{\mathbb{T}}(f;s)=\mathscr{L}_{\mathbb{T}}\{p_{f \boxminus_{\mu} 1}(\cdot,s)\}(1)=\displaystyle\int_0^{\infty} p_{f \boxminus_{\mu}1}(\eta,s) e_{\ominus_{\mu}1}^{\sigma}(\eta,0) \Delta \eta,$$ | |
| − | where $\mathrm{id}$ denotes the identity map $\mathrm{id} \colon \mathbb{T} \rightarrow \mathbb{T}$ and $e_{\ | + | 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]]<br /> | |
| − | With these definitions, we | + | [[Gamma function diverges at zero]]<br /> |
| − | $$\Gamma_{\mathbb{T}}(f;s)=\mathscr{L}_{\mathbb{T}}\{p_{f \boxminus_{\mu} 1}(\cdot,s)\}(1)=\displaystyle\int_0^{\infty} p_{f \boxminus_{\mu}1}(\eta,s) e_{\ominus_{\mu}1}^{\sigma}(\eta,0) \Delta \eta.$$ | + | [[Gamma function diverges at infinity]]<br /> |
| + | [[Gamma function equals one at one]]<br /> | ||
| + | [[Gamma function of x boxplus one]]<br /> | ||
| + | [[Gamma function on certain time scales at bracket number equals bracket factorial]]<br /> | ||
| + | |||
| + | =Examples of gamma functions= | ||
| + | We write formulas for gamma functions defined for $x \in \mathbb{R}^+$ and $s \in \mathbb{T}^+$. | ||
| + | <center> | ||
| + | {| class="wikitable" | ||
| + | |- | ||
| + | |$\mathbb{T}=$ | ||
| + | |$\Gamma_{\mathbb{T}}(x;s)=$ | ||
| + | |- | ||
| + | |[[Real_numbers | $\mathbb{R}$]] | ||
| + | |$\displaystyle\int_0^{\infty} \left( \dfrac{\tau}{s} \right)^{x-1}e^{-\tau} \mathrm{d}\tau$ | ||
| + | |- | ||
| + | |[[Multiples_of_integers | $h\mathbb{Z};h>0$]] | ||
| + | |$h \displaystyle\sum_{k=0}^{\infty} \left( \displaystyle\prod_{j=s}^{k-1} \dfrac{j+x}{j+1} \right) \dfrac{1}{(1+h)^{k+1}}$ | ||
| + | |- | ||
| + | |[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}; q>1$]] | ||
| + | |$\dfrac{(q-1)s}{(1+(q-1)x)^{\log_q(s)}} \displaystyle\sum_{k=-\infty}^{\infty} \dfrac{(1+(q-1)x)^k}{\prod_{j=-\infty}^{k} (1+(q-1)q^k)}$ | ||
| + | |} | ||
| + | </center> | ||
| + | |||
| + | =References= | ||
| + | * {{PaperReference|The gamma function on time scales|2013|Martin Bohner|author2=Başak Karpuz|prev=Laplace transform|next=}}: Definition 2 | ||
| + | |||
| + | [[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
- Martin Bohner and Başak Karpuz: The gamma function on time scales (2013)... (previous): Definition 2
