The q-gamma function and q-binomial coefficients

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The q-gamma function and q-binomial coefficients

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The q-gamma function and q-binomial coefficients

${\displaystyle{\displaystyle{\displaystyle{}\Gamma_{q}\left(x\right):=\frac{\qPochhammer{q}{q}{\infty}}{\qPochhammer{q^{x}}{q}{\infty}}(1-q)^{1-x}}}}$

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

${\displaystyle{\displaystyle{\displaystyle\lim\limits_{q\rightarrow 1}\Gamma_{q}\left(x\right)=\Gamma\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\Gamma_{q}\left(z+1\right)=\frac{1-q^{z}}{1-q}\Gamma_{q}\left(z\right)\quad\textrm{with}\quad\Gamma_{q}\left(1\right)=1}}}$
${\displaystyle{\displaystyle{\displaystyle\Gamma_{q}\left(x\right)=\frac{\qPochhammer{q^{-1}}{q^{-1}}{\infty}}{\qPochhammer{q^{-x}}{q^{-1}}{\infty}}q^{\binomial{x}{2}}(q-1)^{1-x}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle q>1}}}$

${\displaystyle{\displaystyle{\displaystyle{}\qBinomial{n}{k}{q}:=\frac{\qPochhammer{q}{q}{n}}{\qPochhammer{q}{q}{k}\qPochhammer{q}{q}{n-k}}=\qBinomial{n}{n-k}{q}}}}$

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

${\displaystyle{\displaystyle{\displaystyle\qBinomial{\alpha}{k}{q}:=\frac{\left(q^{-\alpha};q\right)_{k}}{\left(q;q\right)_{k}}(-1)^{k}q^{k\alpha-\binomial{k}{2}}}}}$
${\displaystyle{\displaystyle{\displaystyle\qBinomial{\alpha}{\beta}{q}:=\frac{\Gamma_{q}\left(\alpha+1\right)}{\Gamma_{q}\left(\beta+1\right)\Gamma_{q}\left(\alpha-\beta+1\right)}=\frac{\left(q^{\beta+1};q\right)_{\infty}\left(q^{\alpha-\beta+1};q\right)_{\infty}}{\left(q;q\right)_{\infty}\left(q^{\alpha+1};q\right)_{\infty}}}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{\left(q^{\alpha+1};q\right)_{n}}{\left(q;q\right)_{n}}=\qBinomial{n+\alpha}{n}{q}}}}$
${\displaystyle{\displaystyle{\displaystyle\lim\limits_{q\rightarrow 1}\qBinomial{\alpha}{\beta}{q}=\frac{\Gamma\left(\alpha+1\right)}{\Gamma\left(\beta+1\right)\Gamma\left(\alpha-\beta+1\right)}=\binomial{\alpha}{\beta}}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{1}{\qPochhammer{q}{q}{n}}=\sum_{k=0}^{n}\frac{q^{k}}{\qPochhammer{q}{q}{k}}}}}$

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

${\displaystyle{\displaystyle{\displaystyle\qPochhammer{a}{q}{n}=\sum_{k=0}^{n}\qBinomial{n}{k}{q}q^{\binomial{k}{2}}(-a)^{k}}}}$

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

<< The q-shifted factorial

The q-gamma function and q-binomial coefficients

Basic hypergeometric functions >>