Formula:KLS:01.08:19

<< Formula:KLS:01.08:17

formula in The q-shifted factorial

Formula:KLS:01.08:20 >>

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

Constraint(s)

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

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

${\displaystyle{\displaystyle{\displaystyle 0<|q|<1}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{n-k}=\frac{\left(a;q\right)_{n}}{\left(a^{-1}q^{1-n};q\right)_{k}}\left(-\frac{q}{a}\right)^{k}q^{\binomial{k}{2}-nk},\quad a\neq 0}}}$ &

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

${\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{\infty}=\prod_{k=0}^{\infty}(1-aq^{k})=\left(a;q^{2}\right)_{\infty}\left(aq;q^{2}\right)_{\infty}}}}$

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
${\displaystyle{\displaystyle{\displaystyle\Pi}}}$ : product : http://drmf.wmflabs.org/wiki/Definition:prod

Bibliography

Equation in Section 1.8 of KLS.

URL links

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<< Formula:KLS:01.08:17

formula in The q-shifted factorial

Formula:KLS:01.08:20 >>