Formula:KLS:01.09:07

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formula in The q-gamma function and q-binomial coefficients

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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}} }}

Proof

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Symbols List

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \genfrac[]\z@{}{n}{j}_{q}}}  : Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle q}} -binomial coefficient (or Gaussian polynomial) : http://dlmf.nist.gov/17.2#E27 http://dlmf.nist.gov/26.9#SS2.p1
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Gamma_{q}}}  : Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle q}} -gamma function : http://dlmf.nist.gov/5.18#E4
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (a;q)_n}}  : Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle q}} -Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1

Bibliography

Equation in Section 1.9 of KLS.

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formula in The q-gamma function and q-binomial coefficients

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