\qHyperrphis 10 @ @ a - q z = ∑ n = 0 ∞ ( a ; q ) n ( q ; q ) n z n = ( a z ; q ) ∞ ( z ; q ) ∞ , 0 < | q | < 1 formulae-sequence \qHyperrphis 10 @ @ 𝑎 𝑞 𝑧 superscript subscript 𝑛 0 q-Pochhammer-symbol 𝑎 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 superscript 𝑧 𝑛 q-Pochhammer-symbol 𝑎 𝑧 𝑞 q-Pochhammer-symbol 𝑧 𝑞 0 𝑞 1 {\displaystyle{\displaystyle{\displaystyle{}{}\qHyperrphis{1}{0}@@{a}{-}{q}{z}% =\sum_{n=0}^{\infty}\frac{\left(a;q\right)_{n}}{\left(q;q\right)_{n}}z^{n}=% \frac{\left(az;q\right)_{\infty}}{\left(z;q\right)_{\infty}},\quad 0<|q|<1}}} {\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 }
\qHyperrphis 10 @ @ q - n - q z = ( z q - n ; q ) n \qHyperrphis 10 @ @ superscript 𝑞 𝑛 𝑞 𝑧 q-Pochhammer-symbol 𝑧 superscript 𝑞 𝑛 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle\qHyperrphis{1}{0}@@{q^{-n}}{-}{q}{z% }=\left(zq^{-n};q\right)_{n}}}} {\displaystyle \qHyperrphis{1}{0}@@{q^{-n}}{-}{q}{z}=\qPochhammer{zq^{-n}}{q}{n} }
\qHyperrphis 21 @ @ a , b c q c a b = ( a - 1 c , b - 1 c ; q ) ∞ ( c , a - 1 b - 1 c ; q ) ∞ , 0 < | q | < 1 formulae-sequence \qHyperrphis 21 @ @ 𝑎 𝑏 𝑐 𝑞 𝑐 𝑎 𝑏 q-Pochhammer-symbol superscript 𝑎 1 𝑐 superscript 𝑏 1 𝑐 𝑞 q-Pochhammer-symbol 𝑐 superscript 𝑎 1 superscript 𝑏 1 𝑐 𝑞 0 𝑞 1 {\displaystyle{\displaystyle{\displaystyle{}{}\qHyperrphis{2}{1}@@{a,b}{c}{q}{% \frac{c}{ab}}=\frac{\left(a^{-1}c,b^{-1}c;q\right)_{\infty}}{\left(c,a^{-1}b^{% -1}c;q\right)_{\infty}},\quad 0<|q|<1}}} {\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 }
\qHyperrphis 21 @ @ q - n , b c q c q n b = ( b - 1 c ; q ) n ( c ; q ) n \qHyperrphis 21 @ @ superscript 𝑞 𝑛 𝑏 𝑐 𝑞 𝑐 superscript 𝑞 𝑛 𝑏 q-Pochhammer-symbol superscript 𝑏 1 𝑐 𝑞 𝑛 q-Pochhammer-symbol 𝑐 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle{}{}{}{}\qHyperrphis{2}{1}@@{q^{-n},% b}{c}{q}{\frac{cq^{n}}{b}}=\frac{\left(b^{-1}c;q\right)_{n}}{\left(c;q\right)_% {n}}}}} {\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}} }
\qHyperrphis 21 @ @ q - n , b c q q = ( b - 1 c ; q ) n ( c ; q ) n b n \qHyperrphis 21 @ @ superscript 𝑞 𝑛 𝑏 𝑐 𝑞 𝑞 q-Pochhammer-symbol superscript 𝑏 1 𝑐 𝑞 𝑛 q-Pochhammer-symbol 𝑐 𝑞 𝑛 superscript 𝑏 𝑛 {\displaystyle{\displaystyle{\displaystyle\qHyperrphis{2}{1}@@{q^{-n},b}{c}{q}% {q}=\frac{\left(b^{-1}c;q\right)_{n}}{\left(c;q\right)_{n}}b^{n}}}} {\displaystyle \qHyperrphis{2}{1}@@{q^{-n},b}{c}{q}{q}=\frac{\qPochhammer{b^{-1}c}{q}{n}}{\qPochhammer{c}{q}{n}}b^n }
\qHyperrphis 11 @ @ a c q c a = ( a - 1 c ; q ) ∞ ( c ; q ) ∞ \qHyperrphis 11 @ @ 𝑎 𝑐 𝑞 𝑐 𝑎 q-Pochhammer-symbol superscript 𝑎 1 𝑐 𝑞 q-Pochhammer-symbol 𝑐 𝑞 {\displaystyle{\displaystyle{\displaystyle{}\qHyperrphis{1}{1}@@{a}{c}{q}{% \frac{c}{a}}=\frac{\left(a^{-1}c;q\right)_{\infty}}{\left(c;q\right)_{\infty}}% }}} {\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}} }
\qHyperrphis 20 @ @ q - n , b - q q n b = 1 b n \qHyperrphis 20 @ @ superscript 𝑞 𝑛 𝑏 𝑞 superscript 𝑞 𝑛 𝑏 1 superscript 𝑏 𝑛 {\displaystyle{\displaystyle{\displaystyle{}\qHyperrphis{2}{0}@@{q^{-n},b}{-}{% q}{\frac{q^{n}}{b}}=\frac{1}{b^{n}}}}} {\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} }
\qHyperrphis 01 @ @ - c q c = 1 ( c ; q ) ∞ \qHyperrphis 01 @ @ 𝑐 𝑞 𝑐 1 q-Pochhammer-symbol 𝑐 𝑞 {\displaystyle{\displaystyle{\displaystyle{}\qHyperrphis{0}{1}@@{-}{c}{q}{c}=% \frac{1}{\left(c;q\right)_{\infty}}}}} {\displaystyle \index{Summation formula!for a $\qHyperrphis{0}{1}$} \qHyperrphis{0}{1}@@{-}{c}{q}{c}=\frac{1}{\qPochhammer{c}{q}{\infty}} }
\qHyperrphis 32 @ @ q - n , a , b c , a b c - 1 q 1 - n q q = ( a - 1 c , b - 1 c ; q ) n ( c , a - 1 b - 1 c ; q ) n \qHyperrphis 32 @ @ superscript 𝑞 𝑛 𝑎 𝑏 𝑐 𝑎 𝑏 superscript 𝑐 1 superscript 𝑞 1 𝑛 𝑞 𝑞 q-Pochhammer-symbol superscript 𝑎 1 𝑐 superscript 𝑏 1 𝑐 𝑞 𝑛 q-Pochhammer-symbol 𝑐 superscript 𝑎 1 superscript 𝑏 1 𝑐 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle{}{}{}{}\qHyperrphis{3}{2}@@{q^{-n},% a,b}{c,abc^{-1}q^{1-n}}{q}{q}=\frac{\left(a^{-1}c,b^{-1}c;q\right)_{n}}{\left(% c,a^{-1}b^{-1}c;q\right)_{n}}}}} {\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}} }
\qHyperrphis 65 @ @ q a , - q a , a , b , c , d a , - a , a b - 1 q , a c - 1 q , a d - 1 q q a q b c d = ( a q , a b - 1 c - 1 q , a b - 1 d - 1 q , a c - 1 d - 1 q ; q ) ∞ ( a b - 1 q , a c - 1 q , a d - 1 q , a b - 1 c - 1 d - 1 q ; q ) ∞ \qHyperrphis 65 @ @ 𝑞 𝑎 𝑞 𝑎 𝑎 𝑏 𝑐 𝑑 𝑎 𝑎 𝑎 superscript 𝑏 1 𝑞 𝑎 superscript 𝑐 1 𝑞 𝑎 superscript 𝑑 1 𝑞 𝑞 𝑎 𝑞 𝑏 𝑐 𝑑 q-Pochhammer-symbol 𝑎 𝑞 𝑎 superscript 𝑏 1 superscript 𝑐 1 𝑞 𝑎 superscript 𝑏 1 superscript 𝑑 1 𝑞 𝑎 superscript 𝑐 1 superscript 𝑑 1 𝑞 𝑞 q-Pochhammer-symbol 𝑎 superscript 𝑏 1 𝑞 𝑎 superscript 𝑐 1 𝑞 𝑎 superscript 𝑑 1 𝑞 𝑎 superscript 𝑏 1 superscript 𝑐 1 superscript 𝑑 1 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle{}{}{}\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{\left(aq,ab^{-1}c^{-1}q,ab^{-1}d^{-1}q,ac^{-1}d^{-1}q;q% \right)_{\infty}}{\left(ab^{-1}q,ac^{-1}q,ad^{-1}q,ab^{-1}c^{-1}d^{-1}q;q% \right)_{\infty}}}}} {\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}} }
\qHyperrphis 65 @ @ q a , - q a , a , b , c , q - n a , - a , a b - 1 q , a c - 1 q , a q n + 1 q a q n + 1 b c = ( a q , a b - 1 c - 1 q ; q ) n ( a b - 1 q , a c - 1 q ; q ) n \qHyperrphis 65 @ @ 𝑞 𝑎 𝑞 𝑎 𝑎 𝑏 𝑐 superscript 𝑞 𝑛 𝑎 𝑎 𝑎 superscript 𝑏 1 𝑞 𝑎 superscript 𝑐 1 𝑞 𝑎 superscript 𝑞 𝑛 1 𝑞 𝑎 superscript 𝑞 𝑛 1 𝑏 𝑐 q-Pochhammer-symbol 𝑎 𝑞 𝑎 superscript 𝑏 1 superscript 𝑐 1 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑎 superscript 𝑏 1 𝑞 𝑎 superscript 𝑐 1 𝑞 𝑞 𝑛 {\displaystyle{\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{\left(aq,ab^{-1}c^{-1}q;q\right)_{n}}{\left(ab^{-1}q,% ac^{-1}q;q\right)_{n}}}}} {\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}} }
\qHyperrphis 64 @ @ q a , - q a , a , b , 0 , q - n a , - a , a b - 1 q , a q n + 1 q q n b = ( a q ; q ) n ( a b - 1 q ; q ) n b - n \qHyperrphis 64 @ @ 𝑞 𝑎 𝑞 𝑎 𝑎 𝑏 0 superscript 𝑞 𝑛 𝑎 𝑎 𝑎 superscript 𝑏 1 𝑞 𝑎 superscript 𝑞 𝑛 1 𝑞 superscript 𝑞 𝑛 𝑏 q-Pochhammer-symbol 𝑎 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑎 superscript 𝑏 1 𝑞 𝑞 𝑛 superscript 𝑏 𝑛 {\displaystyle{\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{\left(aq;q\right)_{n}}{\left(ab^{-1}q;q\right)_{n}}b^{-n}}}} {\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} }
\qHyperrphis 63 @ @ q a , - q a , a , 0 , 0 , q - n a , - a , a q n + 1 q q n - 1 a = ( - 1 ) n a - n q - \binomial n + 12 ( a q ; q ) n \qHyperrphis 63 @ @ 𝑞 𝑎 𝑞 𝑎 𝑎 0 0 superscript 𝑞 𝑛 𝑎 𝑎 𝑎 superscript 𝑞 𝑛 1 𝑞 superscript 𝑞 𝑛 1 𝑎 superscript 1 𝑛 superscript 𝑎 𝑛 superscript 𝑞 \binomial 𝑛 12 q-Pochhammer-symbol 𝑎 𝑞 𝑞 𝑛 {\displaystyle{\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)^{n}a^{-n}q^{-\binomial{n+1}{2}}\left(aq;q\right)_{n}}}} {\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} }