Formula:KLS:01.10:01

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formula in Basic hypergeometric functions

Formula:KLS:01.10:03 >>

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \index{Basic hypergeometric function} \qHyperrphis{r}{s}@@{a_1,\ldots,a_r}{b_1,\ldots,b_s}{q}{z} {}:=\sum\limits_{k=0}^{\infty}\frac{\qPochhammer{a_1,\ldots,a_r}{q}{k}}{\qPochhammer{b_1,\ldots,b_s}{q}{k}} (-1)^{(1+s-r)k}q^{(1+s-r)\binomial{k}{2}}\frac{z^k}{\qPochhammer{q}{q}{k}} }}

Substitution(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qPochhammer{a_1,\ldots,a_r}{q}{k}:=\qPochhammer{a_1}{q}{k}\cdots\qPochhammer{a_r}{q}{k}}}

Proof

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Symbols List

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {{}_{r}\phi_{s}}}}  : basic hypergeometric (or Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle q}} -hypergeometric) function : http://dlmf.nist.gov/17.4#E1
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Sigma}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
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
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \binom{n}{k}}}  : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1

Bibliography

Equation in Section 1.10 of KLS.

URL links

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<< The q-gamma function and q-binomial coefficients

formula in Basic hypergeometric functions

Formula:KLS:01.10:03 >>