# The q-shifted factorial

## The q-shifted factorial

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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