Formula:KLS:01.08:17

<< Formula:KLS:01.08:14

formula in The q-shifted factorial

Formula:KLS:01.08:19 >>

${\displaystyle{\displaystyle{\displaystyle\frac{\left(aq^{-n};q\right)_{n}}{\left(bq^{-n};q\right)_{n}}=\frac{\left(a^{-1}q;q\right)_{n}}{\left(b^{-1}q;q\right)_{n}}\left(\frac{a}{b}\right)^{n},\quad a\neq 0}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle b\neq 0}}}$ &
${\displaystyle{\displaystyle{\displaystyle a\neq 0}}}$

Substitution(s)

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

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:01.08:14

formula in The q-shifted factorial

Formula:KLS:01.08:19 >>