\HyperpFq 21 @ @ a , b c z = ( 1 - z ) c - a - b \HyperpFq 21 @ @ c - a , c - b c z formulae-sequence \HyperpFq 21 @ @ 𝑎 𝑏 𝑐 𝑧 superscript 1 𝑧 𝑐 𝑎 𝑏 \HyperpFq 21 @ @ 𝑐 𝑎 𝑐 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle{\displaystyle{}{}\HyperpFq{2}{1}@@{a,b}{c}{z}=(1-% z)^{c-a-b}\,\HyperpFq{2}{1}@@{c-a,c-b}{c}{z}}}} {\displaystyle \index{Euler's transformation formula}\index{Transformation formula!Euler} \HyperpFq{2}{1}@@{a,b}{c}{z}=(1-z)^{c-a-b}\,\HyperpFq{2}{1}@@{c-a,c-b}{c}{z} } \HyperpFq 21 @ @ a , b c z = ( 1 - z ) - a \HyperpFq 21 @ @ a , c - b c z z - 1 formulae-sequence \HyperpFq 21 @ @ 𝑎 𝑏 𝑐 𝑧 superscript 1 𝑧 𝑎 \HyperpFq 21 @ @ 𝑎 𝑐 𝑏 𝑐 𝑧 𝑧 1 {\displaystyle{\displaystyle{\displaystyle{}{}\HyperpFq{2}{1}@@{a,b}{c}{z}=(1-% z)^{-a}\,\HyperpFq{2}{1}@@{a,c-b}{c}{\frac{z}{z-1}}}}} {\displaystyle \index{Pfaff-Kummer transformation formula}\index{Transformation formula!Pfaff-Kummer} \HyperpFq{2}{1}@@{a,b}{c}{z}=(1-z)^{-a}\,\HyperpFq{2}{1}@@{a,c-b}{c}{\frac{z}{z-1}} } \HyperpFq 11 @ @ a c z = e z \HyperpFq 11 @ @ c - a c - z \HyperpFq 11 @ @ 𝑎 𝑐 𝑧 𝑧 \HyperpFq 11 @ @ 𝑐 𝑎 𝑐 𝑧 {\displaystyle{\displaystyle{\displaystyle{}{}\HyperpFq{1}{1}@@{a}{c}{z}={% \mathrm{e}^{z}}\,\HyperpFq{1}{1}@@{c-a}{c}{-z}}}} {\displaystyle \index{Kummer's transformation formula}\index{Transformation formula!Kummer} \HyperpFq{1}{1}@@{a}{c}{z}=\expe^z\,\HyperpFq{1}{1}@@{c-a}{c}{-z} } \HyperpFq 11 @ @ - n a x = ( - x ) n ( a ) n \HyperpFq 20 @ @ - n , - a - n + 1 - - 1 x fragments \HyperpFq 11 @ @ n a x superscript 𝑥 𝑛 Pochhammer-symbol 𝑎 𝑛 \HyperpFq 20 @ @ n , a n 1 1 𝑥 {\displaystyle{\displaystyle{\displaystyle\HyperpFq{1}{1}@@{-n}{a}{x}=\frac{(-% x)^{n}}{{\left(a\right)_{n}}}\,\HyperpFq{2}{0}@@{-n,-a-n+1}{-}{-\frac{1}{x}}}}} {\displaystyle \HyperpFq{1}{1}@@{-n}{a}{x}=\frac{(-x)^n}{\pochhammer{a}{n}}\,\HyperpFq{2}{0}@@{-n,-a-n+1}{-}{-\frac{1}{x}} }
\HyperpFq 21 @ @ - n , b c x = ( b ) n ( c ) n ( - x ) n \HyperpFq 21 @ @ - n , - c - n + 1 - b - n + 1 1 x formulae-sequence \HyperpFq 21 @ @ 𝑛 𝑏 𝑐 𝑥 Pochhammer-symbol 𝑏 𝑛 Pochhammer-symbol 𝑐 𝑛 superscript 𝑥 𝑛 \HyperpFq 21 @ @ 𝑛 𝑐 𝑛 1 𝑏 𝑛 1 1 𝑥 {\displaystyle{\displaystyle{\displaystyle\HyperpFq{2}{1}@@{-n,b}{c}{x}=\frac{% {\left(b\right)_{n}}}{{\left(c\right)_{n}}}(-x)^{n}\,\HyperpFq{2}{1}@@{-n,-c-n% +1}{-b-n+1}{\frac{1}{x}}}}} {\displaystyle \HyperpFq{2}{1}@@{-n,b}{c}{x}=\frac{\pochhammer{b}{n}}{\pochhammer{c}{n}}(-x)^n\,\HyperpFq{2}{1}@@{-n,-c-n+1}{-b-n+1}{\frac{1}{x}} }
\HyperpFq 43 @ @ - n , a , b , c d , e , f 1 = ( e - a ) n ( f - a ) n ( e ) n ( f ) n \HyperpFq 43 @ @ - n , a , d - b , d - c d , a - e - n + 1 , a - f - n + 11 formulae-sequence \HyperpFq 43 @ @ 𝑛 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 1 Pochhammer-symbol 𝑒 𝑎 𝑛 Pochhammer-symbol 𝑓 𝑎 𝑛 Pochhammer-symbol 𝑒 𝑛 Pochhammer-symbol 𝑓 𝑛 \HyperpFq 43 @ @ 𝑛 𝑎 𝑑 𝑏 𝑑 𝑐 𝑑 𝑎 𝑒 𝑛 1 𝑎 𝑓 𝑛 11 {\displaystyle{\displaystyle{\displaystyle{}{}\HyperpFq{4}{3}@@{-n,a,b,c}{d,e,% f}{1}{}=\frac{{\left(e-a\right)_{n}}{\left(f-a\right)_{n}}}{{\left(e\right)_{n% }}{\left(f\right)_{n}}}\HyperpFq{4}{3}@@{-n,a,d-b,d-c}{d,a-e-n+1,a-f-n+1}{1}}}} {\displaystyle \index{Whipple's transformation formula}\index{Transformation formula!Whipple} \HyperpFq{4}{3}@@{-n,a,b,c}{d,e,f}{1} {}=\frac{\pochhammer{e-a}{n}\pochhammer{f-a}{n}}{\pochhammer{e}{n}\pochhammer{f}{n}}\HyperpFq{4}{3}@@{-n,a,d-b,d-c}{d,a-e-n+1,a-f-n+1}{1} }