ϕ 0 1 ( a - ; q , z ) = ∑ n = 0 ∞ \qPochhammer a q n \qPochhammer q q n z n = \qPochhammer a z q ∞ \qPochhammer z q ∞ , 0 < | q | < 1 formulae-sequence q-hypergeometric-rphis 1 0 𝑎 𝑞 𝑧 superscript subscript 𝑛 0 \qPochhammer 𝑎 𝑞 𝑛 \qPochhammer 𝑞 𝑞 𝑛 superscript 𝑧 𝑛 \qPochhammer 𝑎 𝑧 𝑞 \qPochhammer 𝑧 𝑞 0 𝑞 1 {\displaystyle{\displaystyle{\displaystyle{}{}{{}_{1}\phi_{0}}\!\left({a\atop-% };q,z\right)=\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}}} {\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 }
ϕ 0 1 ( q - n - ; q , z ) = \qPochhammer z q - n q n q-hypergeometric-rphis 1 0 superscript 𝑞 𝑛 𝑞 𝑧 \qPochhammer 𝑧 superscript 𝑞 𝑛 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle{{}_{1}\phi_{0}}\!\left({q^{-n}\atop% -};q,z\right)=\qPochhammer{zq^{-n}}{q}{n}}}} {\displaystyle \qHyperrphis{1}{0}@@{q^{-n}}{-}{q}{z}=\qPochhammer{zq^{-n}}{q}{n} }
ϕ 1 2 ( a , b c ; q , c a b ) = \qPochhammer a - 1 c , b - 1 c q ∞ \qPochhammer c , a - 1 b - 1 c q ∞ , 0 < | q | < 1 formulae-sequence q-hypergeometric-rphis 2 1 𝑎 𝑏 𝑐 𝑞 𝑐 𝑎 𝑏 \qPochhammer superscript 𝑎 1 𝑐 superscript 𝑏 1 𝑐 𝑞 \qPochhammer 𝑐 superscript 𝑎 1 superscript 𝑏 1 𝑐 𝑞 0 𝑞 1 {\displaystyle{\displaystyle{\displaystyle{}{}{{}_{2}\phi_{1}}\!\left({a,b% \atop c};q,\frac{c}{ab}\right)=\frac{\qPochhammer{a^{-1}c,b^{-1}c}{q}{\infty}}% {\qPochhammer{c,a^{-1}b^{-1}c}{q}{\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 }
ϕ 1 2 ( q - n , b c ; q , c q n b ) = \qPochhammer b - 1 c q n \qPochhammer c q n q-hypergeometric-rphis 2 1 superscript 𝑞 𝑛 𝑏 𝑐 𝑞 𝑐 superscript 𝑞 𝑛 𝑏 \qPochhammer superscript 𝑏 1 𝑐 𝑞 𝑛 \qPochhammer 𝑐 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle{}{}{}{}{{}_{2}\phi_{1}}\!\left({q^{% -n},b\atop c};q,\frac{cq^{n}}{b}\right)=\frac{\qPochhammer{b^{-1}c}{q}{n}}{% \qPochhammer{c}{q}{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}} }
ϕ 1 2 ( q - n , b c ; q , q ) = \qPochhammer b - 1 c q n \qPochhammer c q n b n q-hypergeometric-rphis 2 1 superscript 𝑞 𝑛 𝑏 𝑐 𝑞 𝑞 \qPochhammer superscript 𝑏 1 𝑐 𝑞 𝑛 \qPochhammer 𝑐 𝑞 𝑛 superscript 𝑏 𝑛 {\displaystyle{\displaystyle{\displaystyle{{}_{2}\phi_{1}}\!\left({q^{-n},b% \atop c};q,q\right)=\frac{\qPochhammer{b^{-1}c}{q}{n}}{\qPochhammer{c}{q}{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 }
ϕ 1 1 ( a c ; q , c a ) = \qPochhammer a - 1 c q ∞ \qPochhammer c q ∞ q-hypergeometric-rphis 1 1 𝑎 𝑐 𝑞 𝑐 𝑎 \qPochhammer superscript 𝑎 1 𝑐 𝑞 \qPochhammer 𝑐 𝑞 {\displaystyle{\displaystyle{\displaystyle{}{{}_{1}\phi_{1}}\!\left({a\atop c}% ;q,\frac{c}{a}\right)=\frac{\qPochhammer{a^{-1}c}{q}{\infty}}{\qPochhammer{c}{% q}{\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}} }
ϕ 0 2 ( q - n , b - ; q , q n b ) = 1 b n q-hypergeometric-rphis 2 0 superscript 𝑞 𝑛 𝑏 𝑞 superscript 𝑞 𝑛 𝑏 1 superscript 𝑏 𝑛 {\displaystyle{\displaystyle{\displaystyle{}{{}_{2}\phi_{0}}\!\left({q^{-n},b% \atop-};q,\frac{q^{n}}{b}\right)=\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} }
ϕ 1 0 ( - c ; q , c ) = 1 \qPochhammer c q ∞ q-hypergeometric-rphis 0 1 𝑐 𝑞 𝑐 1 \qPochhammer 𝑐 𝑞 {\displaystyle{\displaystyle{\displaystyle{}{{}_{0}\phi_{1}}\!\left({-\atop c}% ;q,c\right)=\frac{1}{\qPochhammer{c}{q}{\infty}}}}} {\displaystyle \index{Summation formula!for a $\qHyperrphis{0}{1}$} \qHyperrphis{0}{1}@@{-}{c}{q}{c}=\frac{1}{\qPochhammer{c}{q}{\infty}} }
ϕ 2 3 ( q - n , a , b c , a b c - 1 q 1 - n ; q , q ) = \qPochhammer a - 1 c , b - 1 c q n \qPochhammer c , a - 1 b - 1 c q n q-hypergeometric-rphis 3 2 superscript 𝑞 𝑛 𝑎 𝑏 𝑐 𝑎 𝑏 superscript 𝑐 1 superscript 𝑞 1 𝑛 𝑞 𝑞 \qPochhammer superscript 𝑎 1 𝑐 superscript 𝑏 1 𝑐 𝑞 𝑛 \qPochhammer 𝑐 superscript 𝑎 1 superscript 𝑏 1 𝑐 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle{}{}{}{}{{}_{3}\phi_{2}}\!\left({q^{% -n},a,b\atop c,abc^{-1}q^{1-n}};q,q\right)=\frac{\qPochhammer{a^{-1}c,b^{-1}c}% {q}{n}}{\qPochhammer{c,a^{-1}b^{-1}c}{q}{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}} }
ϕ 5 6 ( 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 ) = \qPochhammer a q , a b - 1 c - 1 q , a b - 1 d - 1 q , a c - 1 d - 1 q q ∞ \qPochhammer a b - 1 q , a c - 1 q , a d - 1 q , a b - 1 c - 1 d - 1 q q ∞ q-hypergeometric-rphis 6 5 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 \qPochhammer 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 \qPochhammer 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{}{}{}{{}_{6}\phi_{5}}\!\left({q% \sqrt{a},-q\sqrt{a},a,b,c,d\atop\sqrt{a},-\sqrt{a},ab^{-1}q,ac^{-1}q,ad^{-1}q}% ;q,\frac{aq}{bcd}\right){}=\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}}}}} {\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}} }
ϕ 5 6 ( 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 ) = \qPochhammer a q , a b - 1 c - 1 q q n \qPochhammer a b - 1 q , a c - 1 q q n q-hypergeometric-rphis 6 5 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 \qPochhammer 𝑎 𝑞 𝑎 superscript 𝑏 1 superscript 𝑐 1 𝑞 𝑞 𝑛 \qPochhammer 𝑎 superscript 𝑏 1 𝑞 𝑎 superscript 𝑐 1 𝑞 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle{{}_{6}\phi_{5}}\!\left({q\sqrt{a},-% q\sqrt{a},a,b,c,q^{-n}\atop\sqrt{a},-\sqrt{a},ab^{-1}q,ac^{-1}q,aq^{n+1}};q,% \frac{aq^{n+1}}{bc}\right){}=\frac{\qPochhammer{aq,ab^{-1}c^{-1}q}{q}{n}}{% \qPochhammer{ab^{-1}q,ac^{-1}q}{q}{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}} }
ϕ 4 6 ( q a , - q a , a , b , 0 , q - n a , - a , a b - 1 q , a q n + 1 ; q , q n b ) = \qPochhammer a q q n \qPochhammer a b - 1 q q n b - n q-hypergeometric-rphis 6 4 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 \qPochhammer 𝑎 𝑞 𝑞 𝑛 \qPochhammer 𝑎 superscript 𝑏 1 𝑞 𝑞 𝑛 superscript 𝑏 𝑛 {\displaystyle{\displaystyle{\displaystyle{{}_{6}\phi_{4}}\!\left({q\sqrt{a},-% q\sqrt{a},a,b,0,q^{-n}\atop\sqrt{a},-\sqrt{a},ab^{-1}q,aq^{n+1}};q,\frac{q^{n}% }{b}\right){}=\frac{\qPochhammer{aq}{q}{n}}{\qPochhammer{ab^{-1}q}{q}{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} }
ϕ 3 6 ( 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 \qPochhammer a q q n q-hypergeometric-rphis 6 3 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 \qPochhammer 𝑎 𝑞 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle{{}_{6}\phi_{3}}\!\left({q\sqrt{a},-% q\sqrt{a},a,0,0,q^{-n}\atop\sqrt{a},-\sqrt{a},aq^{n+1}};q,\frac{q^{n-1}}{a}% \right){}=(-1)^{n}a^{-n}q^{-\binomial{n+1}{2}}\qPochhammer{aq}{q}{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} }