Difference between revisions of "Q-Gamma"

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(References)
(References)
 
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Let $0<q<1$. Define the $q$-gamma function by the formula
 
Let $0<q<1$. Define the $q$-gamma function by the formula
 
$$\Gamma_q(z) = \dfrac{(q;q)_{\infty}}{(q^z;q)_{\infty}}(1-q)^{1-z},$$
 
$$\Gamma_q(z) = \dfrac{(q;q)_{\infty}}{(q^z;q)_{\infty}}(1-q)^{1-z},$$
where $(\cdot;\cdot)_{\infty}$ denotes the [[q-factorial]]. The function $\Gamma_q$ is a [[q-analogue | $q$-analogue]] of the [[gamma function]].
+
where $(\cdot;\cdot)_{\infty}$ denotes the [[q-shifted factorial]]. The function $\Gamma_q$ is a [[q-analogue | $q$-analogue]] of the [[gamma function]].
  
 
<div align="center">
 
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=References=
 
=References=
 
* {{PaperReference|The q-Gamma and q-Beta functions|1978|Richard Askey|next=findme}}
 
* {{PaperReference|The q-Gamma and q-Beta functions|1978|Richard Askey|next=findme}}
* {{PaperReference|The q-gamma function for q greater than 1|1980|Daniel S. Moak|next=findme}}
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* {{PaperReference|The q-gamma function for q greater than 1|1980|Daniel S. Moak|next=Q-shifted factorial}}
  
 
{{:q-calculus footer}}
 
{{:q-calculus footer}}
  
 
[[Category:SpecialFunction]]
 
[[Category:SpecialFunction]]

Latest revision as of 00:13, 30 May 2017

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-shifted factorial. The function $\Gamma_q$ is a $q$-analogue of the gamma function.

Properties

q-Gamma at z+1
q-Gamma at 1
q-Gamma at 2

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

$q$-calculus