Formula:KLS:01.08:22

<< Formula:KLS:01.08:21

formula in The q-shifted factorial

Formula:KLS:01.08:23 >>

${\displaystyle{\displaystyle{\displaystyle\left(aq^{-n};q\right)_{n-k}=\frac{\left(a^{-1}q;q\right)_{n}}{\left(a^{-1}q;q\right)_{k}}\left(-\frac{a}{q}\right)^{n-k}q^{\binomial{k}{2}-\binomial{n}{2}}{}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle a\neq 0}}}$ &
${\displaystyle{\displaystyle{\displaystyle k=0,1,2,\ldots,n}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-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 1.8 of KLS.

URL links

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<< Formula:KLS:01.08:21

formula in The q-shifted factorial

Formula:KLS:01.08:23 >>