Formula:KLS:01.09:07

<< Formula:KLS:01.09:06

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

Formula:KLS:01.09:08 >>

${\displaystyle{\displaystyle{\displaystyle\genfrac{[}{]}{0.0pt}{}{\alpha}{\beta}_{q}:=\frac{\Gamma_{q}\left(\alpha+1\right)}{\Gamma_{q}\left(\beta+1\right)\Gamma_{q}\left(\alpha-\beta+1\right)}=\frac{\left(q^{\beta+1};q\right)_{\infty}\left(q^{\alpha-\beta+1};q\right)_{\infty}}{\left(q;q\right)_{\infty}\left(q^{\alpha+1};q\right)_{\infty}}}}}$

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

${\displaystyle{\displaystyle{\displaystyle\genfrac{[}{]}{0.0pt}{0}{}{n}{j}_{q}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-binomial coefficient (or Gaussian polynomial) : http://dlmf.nist.gov/17.2#E27 http://dlmf.nist.gov/26.9#SS2.p1
${\displaystyle{\displaystyle{\displaystyle\Gamma_{q}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-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

Bibliography

Equation in Section 1.9 of KLS.

URL links

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

<< Formula:KLS:01.09:06

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

Formula:KLS:01.09:08 >>