The q-binomial theorem and other summation formulas

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The q-binomial theorem and other summation formulas

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \index{q-Binomial theorem@$q$-Binomial theorem}\index{Summation formula!q-Binomial theorem@$q$-Binomial theorem} \qHyperrphis{1}{0}@@{a}{-}{q}{z}=\sum_{n=0}^{\infty}\frac{\qPochhammer{a}{q}{n}}{\qPochhammer{q}{q}{n}}z^n= \frac{\qPochhammer{az}{q}{\infty}}{\qPochhammer{z}{q}{\infty}},\quad 0<|q|<1 }}

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


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qHyperrphis{1}{0}@@{q^{-n}}{-}{q}{z}=\qPochhammer{zq^{-n}}{q}{n} }}

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 \index{q-Gauss summation formula@$q$-Gauss summation formula}\index{Summation formula!q-Gauss@$q$-Gauss} \qHyperrphis{2}{1}@@{a,b}{c}{q}{\frac{c}{ab}}=\frac{\qPochhammer{a^{-1}c,b^{-1}c}{q}{\infty}} {\qPochhammer{c,a^{-1}b^{-1}c}{q}{\infty}},\quad 0<|q|<1 }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \left|\frac{c}{ab}\right|<1}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \index{q-Vandermonde summation formula@$q$-Vandermonde summation formula}\index{Summation formula!q-Vandermonde@$q$-Vandermonde}\index{q-Chu-Vandermonde summation formula@$q$-Chu-Vandermonde summation formula}\index{Summation formula!q-Chu-Vandermonde@$q$-Chu-Vandermonde} \qHyperrphis{2}{1}@@{q^{-n},b}{c}{q}{\frac{cq^n}{b}}=\frac{\qPochhammer{b^{-1}c}{q}{n}}{\qPochhammer{c}{q}{n}} }}

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 \qHyperrphis{2}{1}@@{q^{-n},b}{c}{q}{q}=\frac{\qPochhammer{b^{-1}c}{q}{n}}{\qPochhammer{c}{q}{n}}b^n }}

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 \index{Summation formula!for a $\qHyperrphis{1}{1}$} \qHyperrphis{1}{1}@@{a}{c}{q}{\frac{c}{a}}=\frac{\qPochhammer{a^{-1}c}{q}{\infty}}{\qPochhammer{c}{q}{\infty}} }}

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 \index{Summation formula!for a terminating $\qHyperrphis{2}{0}$} \qHyperrphis{2}{0}@@{q^{-n},b}{-}{q}{\frac{q^n}{b}}=\frac{1}{b^n} }}

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 \index{Summation formula!for a $\qHyperrphis{0}{1}$} \qHyperrphis{0}{1}@@{-}{c}{q}{c}=\frac{1}{\qPochhammer{c}{q}{\infty}} }}

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 \index{q-Saalschutz summation formula@$q$-Saalsch\"{u}tz summation formula}\index{Summation formula!q-Saalschutz@$q$-Saalsch\"{u}tz}\index{q-Pfaff-Saalschutz summation formula@$q$-Pfaff-Saalsch\"{u}tz summation formula}\index{Summation formula!q-Pfaff-Saalschutz@$q$-Pfaff-Saalsch\"{u}tz} \qHyperrphis{3}{2}@@{q^{-n},a,b}{c,abc^{-1}q^{1-n}}{q}{q}=\frac{\qPochhammer{a^{-1}c,b^{-1}c}{q}{n}} {\qPochhammer{c,a^{-1}b^{-1}c}{q}{n}} }}

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 \index{Jackson's summation formula}\index{Summation formula!Jackson}\index{Summation formula!for a very-well-poised $\qHyperrphis{6}{5}$} \qHyperrphis{6}{5}@@{q\sqrt{a},-q\sqrt{a},a,b,c,d}{\sqrt{a},-\sqrt{a},ab^{-1}q,ac^{-1}q,ad^{-1}q}{q}{\frac{aq}{bcd}} {}=\frac{\qPochhammer{aq,ab^{-1}c^{-1}q,ab^{-1}d^{-1}q,ac^{-1}d^{-1}q}{q}{\infty}} {\qPochhammer{ab^{-1}q,ac^{-1}q,ad^{-1}q,ab^{-1}c^{-1}d^{-1}q}{q}{\infty}} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \left|\frac{aq}{bcd}\right|<1}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qHyperrphis{6}{5}@@{q\sqrt{a},-q\sqrt{a},a,b,c,q^{-n}}{\sqrt{a},-\sqrt{a},ab^{-1}q,ac^{-1}q,aq^{n+1}}{q}{\frac{aq^{n+1}}{bc}} {}=\frac{\qPochhammer{aq,ab^{-1}c^{-1}q}{q}{n}}{\qPochhammer{ab^{-1}q,ac^{-1}q}{q}{n}} }}

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 \qHyperrphis{6}{4}@@{q\sqrt{a},-q\sqrt{a},a,b,0,q^{-n}}{\sqrt{a},-\sqrt{a},ab^{-1}q,aq^{n+1}}{q}{\frac{q^n}{b}} {}=\frac{\qPochhammer{aq}{q}{n}}{\qPochhammer{ab^{-1}q}{q}{n}}b^{-n} }}

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 \qHyperrphis{6}{3}@@{q\sqrt{a},-q\sqrt{a},a,0,0,q^{-n}}{\sqrt{a},-\sqrt{a},aq^{n+1}}{q}{\frac{q^{n-1}}{a}} {}=(-1)^na^{-n}q^{-\binomial{n+1}{2}}\qPochhammer{aq}{q}{n} }}

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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