The q-gamma function and q-binomial coefficients

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The q-gamma function and q-binomial coefficients

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle 0<q<1}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim\limits_{q\rightarrow 1}\qGamma{q}@{x}=\EulerGamma@{x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qGamma{q}@{z+1}=\frac{1-q^z}{1-q}\qGamma{q}@{z}\quad\textrm{with}\quad\qGamma{q}@{1}=1 }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle q>1}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle k=0,1,2,\ldots,n}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qBinomial{\alpha}{k}{q}:=\frac{\qPochhammer{q^{-\alpha}}{q}{k}}{\qPochhammer{q}{q}{k}} (-1)^kq^{k\alpha-\binomial{k}{2}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\qPochhammer{q^{\alpha+1}}{q}{n}}{\qPochhammer{q}{q}{n}}=\qBinomial{n+\alpha}{n}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim\limits_{q\rightarrow 1}\qBinomial{\alpha}{\beta}{q} =\frac{\EulerGamma@{\alpha+1}}{\EulerGamma@{\beta+1}\EulerGamma@{\alpha-\beta+1}}=\binomial{\alpha}{\beta} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{1}{\qPochhammer{q}{q}{n}}=\sum_{k=0}^n\frac{q^k}{\qPochhammer{q}{q}{k}} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle n=0,1,2,\ldots}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qPochhammer{a}{q}{n}=\sum_{k=0}^n\qBinomial{n}{k}{q}q^{\binomial{k}{2}}(-a)^k }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle n=0,1,2,\ldots}}


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