Formula:KLS:01.09:09

<< Formula:KLS:01.09:08

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

Formula:KLS:01.09:10 >>

${\displaystyle{\displaystyle{\displaystyle\lim\limits_{q\rightarrow 1}\genfrac{[}{]}{0.0pt}{}{\alpha}{\beta}_{q}=\frac{\Gamma\left(\alpha+1\right)}{\Gamma\left(\beta+1\right)\Gamma\left(\alpha-\beta+1\right)}=\binomial{\alpha}{\beta}}}}$

Proof

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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
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : 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

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<< Formula:KLS:01.09:08

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

Formula:KLS:01.09:10 >>