Gamma function

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 $\boxminus_{\mu}$ denotes forward box minus, $\ominus_{\mu}$ denotes forward circle minus, and $\sigma$ denotes the forward jump.

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

• Martin Bohner and Başak Karpuz: The gamma function on time scales (2013)... (previous): Definition 2