The q-shifted factorial

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The q-shifted factorial

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The q-shifted factorial

${\displaystyle{\displaystyle{\displaystyle\lim\limits_{q\rightarrow 1}\frac{1-q^{\alpha}}{1-q}=\alpha}}}$
${\displaystyle{\displaystyle{\displaystyle[\alpha]:=\frac{1-q^{\alpha}}{1-q}}}}$
${\displaystyle{\displaystyle{\displaystyle[0]=0,\quad[n]=\frac{1-q^{n}}{1-q}=\sum_{k=0}^{n-1}q^{k}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle n=1,2,3,\ldots}}}$

${\displaystyle{\displaystyle{\displaystyle{}\left(a;q\right)_{0}:=1\quad\textrm{and}\quad\left(a;q\right)_{k}:=\prod_{n=1}^{k}(1-aq^{n-1}),\quad k=1,2,3,\ldots}}}$
${\displaystyle{\displaystyle{\displaystyle\lim\limits_{q\rightarrow 1}\frac{\left(q^{\alpha};q\right)_{k}}{(1-q)^{k}}={\left(\alpha\right)_{k}}}}}$
${\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{-k}:=\frac{1}{\displaystyle\prod_{n=1}^{k}(1-aq^{-n})},\quad a\neq q,q^{2},q^{3},\ldots,q^{k}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle k=1,2,3,\ldots}}}$

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

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots}}}$ &
${\displaystyle{\displaystyle{\displaystyle a\neq 0}}}$

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

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

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle a\neq 0}}}$

${\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{\lambda}=\frac{\left(a;q\right)_{\infty}}{\left(aq^{\lambda};q\right)_{\infty}}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle 0<|q|<1}}}$

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

${\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{n+k}=\left(a;q\right)_{n}\left(aq^{n};q\right)_{k}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle a\neq 0}}}$ &
${\displaystyle{\displaystyle{\displaystyle 0<|q|<1}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{n}=\frac{\left(a;q\right)_{\infty}}{\left(aq^{n};q\right)_{\infty}}=\left(a^{-1}q^{1-n};q\right)_{n}(-a)^{n}q^{\binomial{n}{2}}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{\infty}=\prod_{k=0}^{\infty}(1-aq^{k})=\left(a;q^{2}\right)_{\infty}\left(aq;q^{2}\right)_{\infty}}}}$

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

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle a\neq 0}}}$ &
${\displaystyle{\displaystyle{\displaystyle 0<|q|<1}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{n}=\frac{\left(a;q\right)_{\infty}}{\left(aq^{n};q\right)_{\infty}}=\left(a^{-1}q^{1-n};q\right)_{n}(-a)^{n}q^{\binomial{n}{2}}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{\infty}=\prod_{k=0}^{\infty}(1-aq^{k})=\left(a;q^{2}\right)_{\infty}\left(aq;q^{2}\right)_{\infty}}}}$

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

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle k=0,1,2,\ldots,n}}}$ &

${\displaystyle{\displaystyle{\displaystyle 0<|q|<1}}}$ &
${\displaystyle{\displaystyle{\displaystyle a\neq 0}}}$ &

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

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{n}=\frac{\left(a;q\right)_{\infty}}{\left(aq^{n};q\right)_{\infty}}=\left(a^{-1}q^{1-n};q\right)_{n}(-a)^{n}q^{\binomial{n}{2}}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{\infty}=\prod_{k=0}^{\infty}(1-aq^{k})=\left(a;q^{2}\right)_{\infty}\left(aq;q^{2}\right)_{\infty}}}}$

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

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle b\neq 0}}}$ &
${\displaystyle{\displaystyle{\displaystyle a\neq 0}}}$

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

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

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

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

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

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

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle k=0,1,2,\ldots,n}}}$

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

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle a\neq 0}}}$

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

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle a\neq 0}}}$ &
${\displaystyle{\displaystyle{\displaystyle k=0,1,2,\ldots,n}}}$

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

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle 0<|q|<1}}}$

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

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

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle a\neq 0}}}$ &
${\displaystyle{\displaystyle{\displaystyle 0<|q|<1}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\left(a^{2};q^{2}\right)_{n}=\left(a;q\right)_{n}\left(-a;q\right)_{n}}}}$ &

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

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

<< Transformation formulas

The q-shifted factorial

The q-gamma function and q-binomial coefficients >>