View source for The q-gamma function and q-binomial coefficients

{{DISPLAYTITLE:The ''q''-gamma function and ''q''-binomial coefficients}}
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<div id="alignleft"> << [[The q-shifted factorial|The q-shifted factorial]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#The q-gamma function and q-binomial coefficients|The q-gamma function and q-binomial coefficients]] </div>
<div id="alignright"> [[Basic hypergeometric functions|Basic hypergeometric functions]] >> </div>
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== The ''q''-gamma function and ''q''-binomial coefficients ==

<math id="KLS:01.09:01">{\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}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle 0<q<1}</math></div><br />
<math id="KLS:01.09:02">{\displaystyle 
\lim\limits_{q\rightarrow 1}\qGamma{q}@{x}=\EulerGamma@{x}
}</math><br />
<math id="KLS:01.09:03">{\displaystyle 
\qGamma{q}@{z+1}=\frac{1-q^z}{1-q}\qGamma{q}@{z}\quad\textrm{with}\quad\qGamma{q}@{1}=1
}</math><br />
<math id="KLS:01.09:04">{\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}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle q>1}</math></div><br />
<math id="KLS:01.09:05">{\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}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle k=0,1,2,\ldots,n}</math></div><br />
<math id="KLS:01.09:06">{\displaystyle 
\qBinomial{\alpha}{k}{q}:=\frac{\qPochhammer{q^{-\alpha}}{q}{k}}{\qPochhammer{q}{q}{k}}
(-1)^kq^{k\alpha-\binomial{k}{2}}
}</math><br />
<math id="KLS:01.09:07">{\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}}
}</math><br />
<math id="KLS:01.09:08">{\displaystyle 
\frac{\qPochhammer{q^{\alpha+1}}{q}{n}}{\qPochhammer{q}{q}{n}}=\qBinomial{n+\alpha}{n}{q}
}</math><br />
<math id="KLS:01.09:09">{\displaystyle 
\lim\limits_{q\rightarrow 1}\qBinomial{\alpha}{\beta}{q}
=\frac{\EulerGamma@{\alpha+1}}{\EulerGamma@{\beta+1}\EulerGamma@{\alpha-\beta+1}}=\binomial{\alpha}{\beta}
}</math><br />
<math id="KLS:01.09:10">{\displaystyle 
\frac{1}{\qPochhammer{q}{q}{n}}=\sum_{k=0}^n\frac{q^k}{\qPochhammer{q}{q}{k}}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<math id="KLS:01.09:11">{\displaystyle 
\qPochhammer{a}{q}{n}=\sum_{k=0}^n\qBinomial{n}{k}{q}q^{\binomial{k}{2}}(-a)^k
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<div id="drmf_foot">
<div id="alignleft"> << [[The q-shifted factorial|The q-shifted factorial]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#The q-gamma function and q-binomial coefficients|The q-gamma function and q-binomial coefficients]] </div>
<div id="alignright"> [[Basic hypergeometric functions|Basic hypergeometric functions]] >> </div>
</div>