Formula:KLS:01.09:06

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

Formula:KLS:01.09:07 >>

${\displaystyle{\displaystyle{\displaystyle\genfrac{[}{]}{0.0pt}{}{\alpha}{k}_{q}:=\frac{\left(q^{-\alpha};q\right)_{k}}{\left(q;q\right)_{k}}(-1)^{k}q^{k\alpha-\binomial{k}{2}}}}}$

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
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-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.

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<< Formula:KLS:01.09:05

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

Formula:KLS:01.09:07 >>