Formula:KLS:01.09:11

<< Formula:KLS:01.09:10

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

Formula:KLS:01.10:01 >>

${\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{n}=\sum_{k=0}^{n}\genfrac{[}{]}{0.0pt}{}{n}{k}_{q}q^{\binomial{k}{2}}(-a)^{k}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\genfrac{[}{]}{0.0pt}{0}{}{n}{j}_{q}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-binomial coefficient (or Gaussian polynomial) : http://dlmf.nist.gov/17.2#E27 http://dlmf.nist.gov/26.9#SS2.p1
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : 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:01.09:10

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

Formula:KLS:01.10:01 >>