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

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

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

${\displaystyle{\displaystyle{\displaystyle\qPochhammer{a}{q}{-n}=\frac{1}{\qPochhammer{aq^{-n}}{q}{n}}=\frac{(-qa^{-1})^{n}}{\qPochhammer{qa^{-1}}{q}{n}}q^{\binomial{n}{2}}}}}$

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

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

${\displaystyle{\displaystyle{\displaystyle\qPochhammer{a}{q^{-1}}{n}=\qPochhammer{a^{-1}}{q}{n}(-a)^{n}q^{-\binomial{n}{2}}}}}$

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

${\displaystyle{\displaystyle{\displaystyle\qPochhammer{a}{q}{\lambda}=\frac{\qPochhammer{a}{q}{\infty}}{\qPochhammer{aq^{\lambda}}{q}{\infty}}}}}$

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

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

${\displaystyle{\displaystyle{\displaystyle\qPochhammer{a}{q}{n+k}=\qPochhammer{a}{q}{n}\qPochhammer{aq^{n}}{q}{k}}}}$

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

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

${\displaystyle{\displaystyle{\displaystyle\frac{\qPochhammer{aq^{n}}{q}{k}}{\qPochhammer{aq^{k}}{q}{n}}=\frac{\qPochhammer{a}{q}{k}}{\qPochhammer{a}{q}{n}}}}}$

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

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

${\displaystyle{\displaystyle{\displaystyle\qPochhammer{aq^{k}}{q}{n-k}=\frac{\qPochhammer{a}{q}{n}}{\qPochhammer{a}{q}{k}}}}}$

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): ${\displaystyle{\displaystyle{\displaystyle\qPochhammer{a}{q}{n}=\frac{\qPochhammer{a}{q}{\infty}}{\qPochhammer{aq^{n}}{q}{\infty}}=\qPochhammer{a^{-1}q^{1-n}}{q}{n}(-a)^{n}q^{\binomial{n}{2}}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\qPochhammer{a}{q}{\infty}=\prod_{k=0}^{\infty}(1-aq^{k})=\qPochhammer{a}{q^{2}}{\infty}\qPochhammer{aq}{q^{2}}{\infty}}}}$

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

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

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

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

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

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

${\displaystyle{\displaystyle{\displaystyle\qPochhammer{a}{q}{\infty}=\prod_{k=0}^{\infty}(1-aq^{k})=\qPochhammer{a}{q^{2}}{\infty}\qPochhammer{aq}{q^{2}}{\infty}}}}$

${\displaystyle{\displaystyle{\displaystyle\qPochhammer{q^{-n}}{q}{k}=\frac{\qPochhammer{q}{q}{n}}{\qPochhammer{q}{q}{n-k}}(-1)^{k}q^{\binomial{k}{2}-nk}}}}$

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

${\displaystyle{\displaystyle{\displaystyle\qPochhammer{aq^{-n}}{q}{k}=\frac{\qPochhammer{a^{-1}q}{q}{n}}{\qPochhammer{a^{-1}q^{1-k}}{q}{n}}\qPochhammer{a}{q}{k}q^{-nk}}}}$

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

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

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

${\displaystyle{\displaystyle{\displaystyle\qPochhammer{a}{q}{2n}=\qPochhammer{a}{q^{2}}{n}\qPochhammer{aq}{q^{2}}{n}}}}$
${\displaystyle{\displaystyle{\displaystyle\qPochhammer{a^{2}}{q^{2}}{\infty}=\qPochhammer{a}{q}{\infty}\qPochhammer{-a}{q}{\infty}}}}$

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

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

${\displaystyle{\displaystyle{\displaystyle\frac{1-a^{2}q^{2n}}{1-a^{2}}=\frac{\qPochhammer{a^{2}q^{2}}{q^{2}}{n}}{\qPochhammer{a^{2}}{q^{2}}{n}}=\frac{\qPochhammer{aq}{q}{n}\qPochhammer{-aq}{q}{n}}{\qPochhammer{a}{q}{n}\qPochhammer{-a}{q}{n}}}}}$

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

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle\qPochhammer{a^{2}}{q^{2}}{n}=\qPochhammer{a}{q}{n}\qPochhammer{-a}{q}{n}}}}$ &

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

${\displaystyle{\displaystyle{\displaystyle\qPochhammer{a}{q}{\infty}=\prod_{k=0}^{\infty}(1-aq^{k})=\qPochhammer{a}{q^{2}}{\infty}\qPochhammer{aq}{q^{2}}{\infty}}}}$

<< Transformation formulas

The q-shifted factorial

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