The q-gamma function and q-binomial coefficients: Difference between revisions

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Latest revision as of 00:34, 6 March 2017

The q-gamma function and q-binomial coefficients

Γ q ⁡ ( x ) := ( q ; q ) ∞ ( q x ; q ) ∞ ⁢ ( 1 - q ) 1 - x assign q-Gamma 𝑞 𝑥 q-Pochhammer-symbol 𝑞 𝑞 q-Pochhammer-symbol superscript 𝑞 𝑥 𝑞 superscript 1 𝑞 1 𝑥 {\displaystyle{\displaystyle{\displaystyle{}\Gamma_{q}\left(x\right):=\frac{% \left(q;q\right)_{\infty}}{\left(q^{x};q\right)_{\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} }

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


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 ) = ( q - 1 ; q - 1 ) ∞ ( q - x ; q - 1 ) ∞ ⁢ q \binomial ⁢ x ⁢ 2 ⁢ ( q - 1 ) 1 - x q-Gamma 𝑞 𝑥 q-Pochhammer-symbol superscript 𝑞 1 superscript 𝑞 1 q-Pochhammer-symbol 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{\left% (q^{-1};q^{-1}\right)_{\infty}}{\left(q^{-x};q^{-1}\right)_{\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} }

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


[ n k ] q := ( q ; q ) n ( q ; q ) k ⁢ ( q ; q ) n - k = [ n n - k ] q assign q-binomial 𝑛 𝑘 𝑞 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑘 q-Pochhammer-symbol 𝑞 𝑞 𝑛 𝑘 q-binomial 𝑛 𝑛 𝑘 𝑞 {\displaystyle{\displaystyle{\displaystyle{}\genfrac{[}{]}{0.0pt}{}{n}{k}_{q}:% =\frac{\left(q;q\right)_{n}}{\left(q;q\right)_{k}\left(q;q\right)_{n-k}}=% \genfrac{[}{]}{0.0pt}{}{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} }

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


[ α k ] q := ( q - α ; q ) k ( q ; q ) k ⁢ ( - 1 ) k ⁢ q k ⁢ α - \binomial ⁢ k ⁢ 2 assign q-binomial 𝛼 𝑘 𝑞 q-Pochhammer-symbol superscript 𝑞 𝛼 𝑞 𝑘 q-Pochhammer-symbol 𝑞 𝑞 𝑘 superscript 1 𝑘 superscript 𝑞 𝑘 𝛼 \binomial 𝑘 2 {\displaystyle{\displaystyle{\displaystyle\genfrac{[}{]}{0.0pt}{}{\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}} }
[ 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 q-binomial 𝛼 𝛽 𝑞 q-Gamma 𝑞 𝛼 1 q-Gamma 𝑞 𝛽 1 q-Gamma 𝑞 𝛼 𝛽 1 q-Pochhammer-symbol superscript 𝑞 𝛽 1 𝑞 q-Pochhammer-symbol superscript 𝑞 𝛼 𝛽 1 𝑞 q-Pochhammer-symbol 𝑞 𝑞 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 {\displaystyle{\displaystyle{\displaystyle\genfrac{[}{]}{0.0pt}{}{\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 = [ n + α n ] q q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 q-binomial 𝑛 𝛼 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle\frac{\left(q^{\alpha+1};q\right)_{n% }}{\left(q;q\right)_{n}}=\genfrac{[}{]}{0.0pt}{}{n+\alpha}{n}_{q}}}} {\displaystyle \frac{\qPochhammer{q^{\alpha+1}}{q}{n}}{\qPochhammer{q}{q}{n}}=\qBinomial{n+\alpha}{n}{q} }
lim q → 1 ⁡ [ α β ] q = Γ ⁡ ( α + 1 ) Γ ⁡ ( β + 1 ) ⁢ Γ ⁡ ( α - β + 1 ) = \binomial ⁢ α ⁢ β subscript → 𝑞 1 q-binomial 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}\genfrac% {[}{]}{0.0pt}{}{\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 ( q ; q ) n = ∑ k = 0 n q k ( q ; q ) k 1 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript subscript 𝑘 0 𝑛 superscript 𝑞 𝑘 q-Pochhammer-symbol 𝑞 𝑞 𝑘 {\displaystyle{\displaystyle{\displaystyle\frac{1}{\left(q;q\right)_{n}}=\sum_% {k=0}^{n}\frac{q^{k}}{\left(q;q\right)_{k}}}}} {\displaystyle \frac{1}{\qPochhammer{q}{q}{n}}=\sum_{k=0}^n\frac{q^k}{\qPochhammer{q}{q}{k}} }

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


( a ; q ) n = ∑ k = 0 n [ n k ] q ⁢ q \binomial ⁢ k ⁢ 2 ⁢ ( - a ) k q-Pochhammer-symbol 𝑎 𝑞 𝑛 superscript subscript 𝑘 0 𝑛 q-binomial 𝑛 𝑘 𝑞 superscript 𝑞 \binomial 𝑘 2 superscript 𝑎 𝑘 {\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{n}=\sum_{k=0}^{n}% \genfrac{[}{]}{0.0pt}{}{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 }

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


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