Difference between revisions of "Q-Gamma"
(→Properties) |
|||
| Line 21: | Line 21: | ||
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| − | <strong>Theorem ($q$- | + | <strong>Theorem ($q$-Bohr-Mollerup):</strong> Let $f$ be a function which satisfies |
$$f(x+1) = \dfrac{1-q^x}{1-q}f(x)$$ | $$f(x+1) = \dfrac{1-q^x}{1-q}f(x)$$ | ||
for some $q \in (0,1)$, | for some $q \in (0,1)$, | ||
| Line 29: | Line 29: | ||
Then $f(x) = \Gamma_q(x)$. | Then $f(x) = \Gamma_q(x)$. | ||
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
| − | <strong>Proof:</strong> | + | <strong>Proof:</strong> █ |
</div> | </div> | ||
</div> | </div> | ||
Revision as of 08:49, 16 May 2016
Let $0<q<1$. Define the $q$-gamma function by the formula $$\Gamma_q(z) = \dfrac{(q;q)_{\infty}}{(q^z;q)_{\infty}}(1-q)^{1-z},$$ where $(\cdot;\cdot)_{\infty}$ denotes the q-Pochhammer symbol. The function $\Gamma_q$ is a $q$-analogue of the gamma function.
- Qgamma.png
Graph of $\Gamma_q$ on $[-3,3]$.
Domain coloring of $\Gamma_2$.
Properties
Theorem
The following formula holds: $$\Gamma_q(z+1)=\dfrac{1-q^z}{1-q}\Gamma_q(z),$$ where $\Gamma_q$ denotes the $q$-gamma function and $[z]_q$ denotes the $q$-number of $z$.
Proof
References
Proposition: $\Gamma_q(1)=\Gamma_q(2)=1$
Proof: proof goes here █
Theorem ($q$-Bohr-Mollerup): Let $f$ be a function which satisfies $$f(x+1) = \dfrac{1-q^x}{1-q}f(x)$$ for some $q \in (0,1)$, $$f(1)=1,$$ and $\log f(x)$ is convex for $x>0$. Then $f(x) = \Gamma_q(x)$.
Proof: █
Theorem (Legendre Duplication Formula): $\Gamma_q(2x)\Gamma_{q^2}\left(\dfrac{1}{2}\right)=\Gamma_{q^2}(x)\Gamma_{q^2}\left( x +\dfrac{1}{2} \right)(1+q)^{2x+1}$
Proof: proof goes here █
Theorem: ($q$-analog) The following formula holds: $$\displaystyle\lim_{q \rightarrow 1^-} \Gamma_q(z) = \Gamma(z),$$ where $\Gamma_q$ is the q-Gamma function and $\Gamma$ is the gamma function.
Proof: █
References
Askey, Richard . The q-gamma and q-beta functions. Applicable Anal. 8 (1978/79), no. 2, 125--141.
DLMF entry on q-Gamma and q-Beta functions
