Γ q ( x ) := \qPochhammer q q ∞ \qPochhammer q x q ∞ ( 1 - q ) 1 - x assign q-Gamma 𝑞 𝑥 \qPochhammer 𝑞 𝑞 \qPochhammer superscript 𝑞 𝑥 𝑞 superscript 1 𝑞 1 𝑥 {\displaystyle{\displaystyle{\displaystyle{}\Gamma_{q}\left(x\right):=\frac{% \qPochhammer{q}{q}{\infty}}{\qPochhammer{q^{x}}{q}{\infty}}(1-q)^{1-x}}}} {\displaystyle \index{q-Gamma function@$q$-Gamma function} \qGamma{q}@{x}:=\frac{\qPochhammer{q}{q}{\infty}}{\qPochhammer{q^x}{q}{\infty}}(1-q)^{1-x} }
lim q → 1 Γ q ( x ) = Γ ( x ) subscript → 𝑞 1 q-Gamma 𝑞 𝑥 Euler-Gamma 𝑥 {\displaystyle{\displaystyle{\displaystyle\lim\limits_{q\rightarrow 1}\Gamma_{% q}\left(x\right)=\Gamma\left(x\right)}}} {\displaystyle \lim\limits_{q\rightarrow 1}\qGamma{q}@{x}=\EulerGamma@{x} } Γ q ( z + 1 ) = 1 - q z 1 - q Γ q ( z ) with Γ q ( 1 ) = 1 formulae-sequence q-Gamma 𝑞 𝑧 1 1 superscript 𝑞 𝑧 1 𝑞 q-Gamma 𝑞 𝑧 with q-Gamma 𝑞 1 1 {\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 \qGamma{q}@{z+1}=\frac{1-q^z}{1-q}\qGamma{q}@{z}\quad\textrm{with}\quad\qGamma{q}@{1}=1 } Γ q ( x ) = \qPochhammer q - 1 q - 1 ∞ \qPochhammer q - x q - 1 ∞ q \binomial x 2 ( q - 1 ) 1 - x q-Gamma 𝑞 𝑥 \qPochhammer superscript 𝑞 1 superscript 𝑞 1 \qPochhammer superscript 𝑞 𝑥 superscript 𝑞 1 superscript 𝑞 \binomial 𝑥 2 superscript 𝑞 1 1 𝑥 {\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}}}} {\displaystyle \qGamma{q}@{x}=\frac{\qPochhammer{q^{-1}}{q^{-1}}{\infty}}{\qPochhammer{q^{-x}}{q^{-1}}{\infty}} q^{\binomial{x}{2}}(q-1)^{1-x} }
\qBinomial n k q := \qPochhammer q q n \qPochhammer q q k \qPochhammer q q n - k = \qBinomial n n - k q assign \qBinomial 𝑛 𝑘 𝑞 \qPochhammer 𝑞 𝑞 𝑛 \qPochhammer 𝑞 𝑞 𝑘 \qPochhammer 𝑞 𝑞 𝑛 𝑘 \qBinomial 𝑛 𝑛 𝑘 𝑞 {\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}}}} {\displaystyle \index{q-Binomial coefficient@$q$-Binomial coefficient} \qBinomial{n}{k}{q}:=\frac{\qPochhammer{q}{q}{n}}{\qPochhammer{q}{q}{k}\qPochhammer{q}{q}{n-k}}=\qBinomial{n}{n-k}{q} }
\qBinomial α k q := ( q - α ; q ) k ( q ; q ) k ( - 1 ) k q k α - \binomial k 2 assign \qBinomial 𝛼 𝑘 𝑞 q-Pochhammer superscript 𝑞 𝛼 𝑞 𝑘 q-Pochhammer 𝑞 𝑞 𝑘 superscript 1 𝑘 superscript 𝑞 𝑘 𝛼 \binomial 𝑘 2 {\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 \qBinomial{\alpha}{k}{q}:=\frac{\qPochhammer{q^{-\alpha}}{q}{k}}{\qPochhammer{q}{q}{k}} (-1)^kq^{k\alpha-\binomial{k}{2}} } \qBinomial α β q := Γ q ( α + 1 ) Γ q ( β + 1 ) Γ q ( α - β + 1 ) = ( q β + 1 ; q ) ∞ ( q α - β + 1 ; q ) ∞ ( q ; q ) ∞ ( q α + 1 ; q ) ∞ assign \qBinomial 𝛼 𝛽 𝑞 q-Gamma 𝑞 𝛼 1 q-Gamma 𝑞 𝛽 1 q-Gamma 𝑞 𝛼 𝛽 1 q-Pochhammer superscript 𝑞 𝛽 1 𝑞 q-Pochhammer superscript 𝑞 𝛼 𝛽 1 𝑞 q-Pochhammer 𝑞 𝑞 q-Pochhammer superscript 𝑞 𝛼 1 𝑞 {\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 \qBinomial{\alpha}{\beta}{q}:=\frac{\qGamma{q}@{\alpha+1}}{\qGamma{q}@{\beta+1}\qGamma{q}@{\alpha-\beta+1}} =\frac{\qPochhammer{q^{\beta+1}}{q}{\infty}\qPochhammer{q^{\alpha-\beta+1}}{q}{\infty}} {\qPochhammer{q}{q}{\infty}\qPochhammer{q^{\alpha+1}}{q}{\infty}} } ( q α + 1 ; q ) n ( q ; q ) n = \qBinomial n + α n q q-Pochhammer superscript 𝑞 𝛼 1 𝑞 𝑛 q-Pochhammer 𝑞 𝑞 𝑛 \qBinomial 𝑛 𝛼 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle\frac{\left(q^{\alpha+1};q\right)_{n% }}{\left(q;q\right)_{n}}=\qBinomial{n+\alpha}{n}{q}}}} {\displaystyle \frac{\qPochhammer{q^{\alpha+1}}{q}{n}}{\qPochhammer{q}{q}{n}}=\qBinomial{n+\alpha}{n}{q} } lim q → 1 \qBinomial α β q = Γ ( α + 1 ) Γ ( β + 1 ) Γ ( α - β + 1 ) = \binomial α β subscript → 𝑞 1 \qBinomial 𝛼 𝛽 𝑞 Euler-Gamma 𝛼 1 Euler-Gamma 𝛽 1 Euler-Gamma 𝛼 𝛽 1 \binomial 𝛼 𝛽 {\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 \lim\limits_{q\rightarrow 1}\qBinomial{\alpha}{\beta}{q} =\frac{\EulerGamma@{\alpha+1}}{\EulerGamma@{\beta+1}\EulerGamma@{\alpha-\beta+1}}=\binomial{\alpha}{\beta} } 1 \qPochhammer q q n = ∑ k = 0 n q k \qPochhammer q q k 1 \qPochhammer 𝑞 𝑞 𝑛 superscript subscript 𝑘 0 𝑛 superscript 𝑞 𝑘 \qPochhammer 𝑞 𝑞 𝑘 {\displaystyle{\displaystyle{\displaystyle\frac{1}{\qPochhammer{q}{q}{n}}=\sum% _{k=0}^{n}\frac{q^{k}}{\qPochhammer{q}{q}{k}}}}} {\displaystyle \frac{1}{\qPochhammer{q}{q}{n}}=\sum_{k=0}^n\frac{q^k}{\qPochhammer{q}{q}{k}} }
\qPochhammer a q n = ∑ k = 0 n \qBinomial n k q q \binomial k 2 ( - a ) k \qPochhammer 𝑎 𝑞 𝑛 superscript subscript 𝑘 0 𝑛 \qBinomial 𝑛 𝑘 𝑞 superscript 𝑞 \binomial 𝑘 2 superscript 𝑎 𝑘 {\displaystyle{\displaystyle{\displaystyle\qPochhammer{a}{q}{n}=\sum_{k=0}^{n}% \qBinomial{n}{k}{q}q^{\binomial{k}{2}}(-a)^{k}}}} {\displaystyle \qPochhammer{a}{q}{n}=\sum_{k=0}^n\qBinomial{n}{k}{q}q^{\binomial{k}{2}}(-a)^k }