F 1 2 β‘ ( a , b c ; z ) = ( 1 - z ) c - a - b β’ F 1 2 β‘ ( c - a , c - b c ; z ) Gauss-hypergeometric-pFq 2 1 π π π π§ superscript 1 π§ π π π Gauss-hypergeometric-pFq 2 1 π π π π π π§ {\displaystyle{\displaystyle{\displaystyle{}{}{{}_{2}F_{1}}\!\left({a,b\atop c% };z\right)=(1-z)^{c-a-b}\,{{}_{2}F_{1}}\!\left({c-a,c-b\atop c};z\right)}}} {\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} } F 1 2 β‘ ( a , b c ; z ) = ( 1 - z ) - a β’ F 1 2 β‘ ( a , c - b c ; z z - 1 ) Gauss-hypergeometric-pFq 2 1 π π π π§ superscript 1 π§ π Gauss-hypergeometric-pFq 2 1 π π π π π§ π§ 1 {\displaystyle{\displaystyle{\displaystyle{}{}{{}_{2}F_{1}}\!\left({a,b\atop c% };z\right)=(1-z)^{-a}\,{{}_{2}F_{1}}\!\left({a,c-b\atop c};\frac{z}{z-1}\right% )}}} {\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}} } F 1 1 β‘ ( a c ; z ) = \expe z β’ F 1 1 β‘ ( c - a c ; - z ) Gauss-hypergeometric-pFq 1 1 π π π§ superscript \expe π§ Gauss-hypergeometric-pFq 1 1 π π π π§ {\displaystyle{\displaystyle{\displaystyle{}{}{{}_{1}F_{1}}\!\left({a\atop c};% z\right)=\expe^{z}\,{{}_{1}F_{1}}\!\left({c-a\atop c};-z\right)}}} {\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} } F 1 1 β‘ ( - n a ; x ) = ( - x ) n \pochhammer β’ a β’ n β’ F 0 2 β‘ ( - n , - a - n + 1 - ; - 1 x ) Gauss-hypergeometric-pFq 1 1 π π π₯ superscript π₯ π \pochhammer π π Gauss-hypergeometric-pFq 2 0 π π π 1 1 π₯ {\displaystyle{\displaystyle{\displaystyle{{}_{1}F_{1}}\!\left({-n\atop a};x% \right)=\frac{(-x)^{n}}{\pochhammer{a}{n}}\,{{}_{2}F_{0}}\!\left({-n,-a-n+1% \atop-};-\frac{1}{x}\right)}}} {\displaystyle \HyperpFq{1}{1}@@{-n}{a}{x}=\frac{(-x)^n}{\pochhammer{a}{n}}\,\HyperpFq{2}{0}@@{-n,-a-n+1}{-}{-\frac{1}{x}} }
F 1 2 β‘ ( - n , b c ; x ) = \pochhammer β’ b β’ n \pochhammer β’ c β’ n β’ ( - x ) n β’ F 1 2 β‘ ( - n , - c - n + 1 - b - n + 1 ; 1 x ) Gauss-hypergeometric-pFq 2 1 π π π π₯ \pochhammer π π \pochhammer π π superscript π₯ π Gauss-hypergeometric-pFq 2 1 π π π 1 π π 1 1 π₯ {\displaystyle{\displaystyle{\displaystyle{{}_{2}F_{1}}\!\left({-n,b\atop c};x% \right)=\frac{\pochhammer{b}{n}}{\pochhammer{c}{n}}(-x)^{n}\,{{}_{2}F_{1}}\!% \left({-n,-c-n+1\atop-b-n+1};\frac{1}{x}\right)}}} {\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}} }
F 3 4 β‘ ( - n , a , b , c d , e , f ; 1 ) = \pochhammer β’ e - a β’ n β’ \pochhammer β’ f - a β’ n \pochhammer β’ e β’ n β’ \pochhammer β’ f β’ n β’ F 3 4 β‘ ( - n , a , d - b , d - c d , a - e - n + 1 , a - f - n + 1 ; 1 ) Gauss-hypergeometric-pFq 4 3 π π π π π π π 1 \pochhammer π π π \pochhammer π π π \pochhammer π π \pochhammer π π Gauss-hypergeometric-pFq 4 3 π π π π π π π π π π 1 π π π 1 1 {\displaystyle{\displaystyle{\displaystyle{}{}{{}_{4}F_{3}}\!\left({-n,a,b,c% \atop d,e,f};1\right){}=\frac{\pochhammer{e-a}{n}\pochhammer{f-a}{n}}{% \pochhammer{e}{n}\pochhammer{f}{n}}{{}_{4}F_{3}}\!\left({-n,a,d-b,d-c\atop d,a% -e-n+1,a-f-n+1};1\right)}}} {\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} }