Formula:KLS:01.09:04

<< Formula:KLS:01.09:03

formula in The q-gamma function and q-binomial coefficients

Formula:KLS:01.09:05 >>

${\displaystyle{\displaystyle{\displaystyle\Gamma_{q}\left(x\right)=\frac{\left(q^{-1};q^{-1}\right)_{\infty}}{\left(q^{-x};q^{-1}\right)_{\infty}}q^{\binomial{x}{2}}(q-1)^{1-x}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle q>1}}}$

Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

Symbols List

${\displaystyle{\displaystyle{\displaystyle\Gamma_{q}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-gamma function : http://dlmf.nist.gov/5.18#E4
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1

Bibliography

Equation in Section 1.9 of KLS.

URL links

We ask users to provide relevant URL links in this space.

<< Formula:KLS:01.09:03

formula in The q-gamma function and q-binomial coefficients

Formula:KLS:01.09:05 >>