\HyperpFq 10 @ @ a - z = ∑ n = 0 ∞ ( a ) n n ! z n = ( 1 - z ) - a \HyperpFq 10 @ @ 𝑎 𝑧 superscript subscript 𝑛 0 Pochhammer-symbol 𝑎 𝑛 𝑛 superscript 𝑧 𝑛 superscript 1 𝑧 𝑎 {\displaystyle{\displaystyle{\displaystyle{}{}\HyperpFq{1}{0}@@{a}{-}{z}=\sum_% {n=0}^{\infty}\frac{{\left(a\right)_{n}}}{n!}z^{n}=(1-z)^{-a}}}} {\displaystyle \index{Binomial theorem}\index{Summation formula!Binomial theorem} \HyperpFq{1}{0}@@{a}{-}{z}=\sum_{n=0}^{\infty}\frac{\pochhammer{a}{n}}{n!}z^n=(1-z)^{-a} }
\HyperpFq 10 @ @ - n - z = ∑ k = 0 n ( - n ) k k ! z k = ∑ k = 0 n \binomial n k ( - z ) k = ( 1 - z ) n \HyperpFq 10 @ @ 𝑛 𝑧 superscript subscript 𝑘 0 𝑛 Pochhammer-symbol 𝑛 𝑘 𝑘 superscript 𝑧 𝑘 superscript subscript 𝑘 0 𝑛 \binomial 𝑛 𝑘 superscript 𝑧 𝑘 superscript 1 𝑧 𝑛 {\displaystyle{\displaystyle{\displaystyle\HyperpFq{1}{0}@@{-n}{-}{z}=\sum_{k=% 0}^{n}\frac{{\left(-n\right)_{k}}}{k!}z^{k}=\sum_{k=0}^{n}\binomial{n}{k}(-z)^% {k}=(1-z)^{n}}}} {\displaystyle \HyperpFq{1}{0}@@{-n}{-}{z}=\sum_{k=0}^n\frac{\pochhammer{-n}{k}}{k!}z^k =\sum_{k=0}^n\binomial{n}{k}(-z)^k=(1-z)^n }
\HyperpFq 21 @ @ a , b c 1 = Γ ( c ) Γ ( c - a - b ) Γ ( c - a ) Γ ( c - b ) \HyperpFq 21 @ @ 𝑎 𝑏 𝑐 1 Euler-Gamma 𝑐 Euler-Gamma 𝑐 𝑎 𝑏 Euler-Gamma 𝑐 𝑎 Euler-Gamma 𝑐 𝑏 {\displaystyle{\displaystyle{\displaystyle{}{}\HyperpFq{2}{1}@@{a,b}{c}{1}=% \frac{\Gamma\left(c\right)\Gamma\left(c-a-b\right)}{\Gamma\left(c-a\right)% \Gamma\left(c-b\right)}}}} {\displaystyle \index{Gauss summation formula}\index{Summation formula!Gauss} \HyperpFq{2}{1}@@{a,b}{c}{1}= \frac{\EulerGamma@{c}\EulerGamma@{c-a-b}}{\EulerGamma@{c-a}\EulerGamma@{c-b}} }
\HyperpFq 21 @ @ - n , b c 1 = ( c - b ) n ( c ) n \HyperpFq 21 @ @ 𝑛 𝑏 𝑐 1 Pochhammer-symbol 𝑐 𝑏 𝑛 Pochhammer-symbol 𝑐 𝑛 {\displaystyle{\displaystyle{\displaystyle{}{}{}{}\HyperpFq{2}{1}@@{-n,b}{c}{1% }=\frac{{\left(c-b\right)_{n}}}{{\left(c\right)_{n}}}}}} {\displaystyle \index{Vandermonde summation formula}\index{Summation formula!Vandermonde}\index{Chu-Vandermonde summation formula}\index{Summation formula!Chu-Vandermonde} \HyperpFq{2}{1}@@{-n,b}{c}{1}=\frac{\pochhammer{c-b}{n}}{\pochhammer{c}{n}} }
\HyperpFq 32 @ @ - n , a , b c , 1 + a + b - c - n 1 = ( c - a ) n ( c - b ) n ( c ) n ( c - a - b ) n \HyperpFq 32 @ @ 𝑛 𝑎 𝑏 𝑐 1 𝑎 𝑏 𝑐 𝑛 1 Pochhammer-symbol 𝑐 𝑎 𝑛 Pochhammer-symbol 𝑐 𝑏 𝑛 Pochhammer-symbol 𝑐 𝑛 Pochhammer-symbol 𝑐 𝑎 𝑏 𝑛 {\displaystyle{\displaystyle{\displaystyle{}{}{}{}\HyperpFq{3}{2}@@{-n,a,b}{c,% 1+a+b-c-n}{1}=\frac{{\left(c-a\right)_{n}}{\left(c-b\right)_{n}}}{{\left(c% \right)_{n}}{\left(c-a-b\right)_{n}}}}}} {\displaystyle \index{Saalsch\"{u}tz summation formula}\index{Summation formula!Saalsch\"{u}tz}\index{Pfaff-Saalsch\"{u}tz summation formula}\index{Summation formula!Pfaff-Saalsch\"{u}tz} \HyperpFq{3}{2}@@{-n,a,b}{c,1+a+b-c-n}{1}=\frac{\pochhammer{c-a}{n}\pochhammer{c-b}{n}}{\pochhammer{c}{n}\pochhammer{c-a-b}{n}} }
\HyperpFq 54 @ @ 1 + a / 2 , a , b , c , d a / 2 , 1 + a - b , 1 + a - c , 1 + a - d 1 = Γ ( 1 + a - b ) Γ ( 1 + a - c ) Γ ( 1 + a - d ) Γ ( 1 + a - b - c - d ) Γ ( 1 + a ) Γ ( 1 + a - b - c ) Γ ( 1 + a - b - d ) Γ ( 1 + a - c - d ) \HyperpFq 54 @ @ 1 𝑎 2 𝑎 𝑏 𝑐 𝑑 𝑎 2 1 𝑎 𝑏 1 𝑎 𝑐 1 𝑎 𝑑 1 Euler-Gamma 1 𝑎 𝑏 Euler-Gamma 1 𝑎 𝑐 Euler-Gamma 1 𝑎 𝑑 Euler-Gamma 1 𝑎 𝑏 𝑐 𝑑 Euler-Gamma 1 𝑎 Euler-Gamma 1 𝑎 𝑏 𝑐 Euler-Gamma 1 𝑎 𝑏 𝑑 Euler-Gamma 1 𝑎 𝑐 𝑑 {\displaystyle{\displaystyle{\displaystyle{}\HyperpFq{5}{4}@@{1+a/2,a,b,c,d}{a% /2,1+a-b,1+a-c,1+a-d}{1}{}=\frac{\Gamma\left(1+a-b\right)\Gamma\left(1+a-c% \right)\Gamma\left(1+a-d\right)\Gamma\left(1+a-b-c-d\right)}{\Gamma\left(1+a% \right)\Gamma\left(1+a-b-c\right)\Gamma\left(1+a-b-d\right)\Gamma\left(1+a-c-d% \right)}}}} {\displaystyle \index{Summation formula!for a very-well-poised $\HyperpFq{5}{4}$} \HyperpFq{5}{4}@@{1+a/2,a,b,c,d}{a/2,1+a-b,1+a-c,1+a-d}{1} {}=\frac{\EulerGamma@{1+a-b}\EulerGamma@{1+a-c}\EulerGamma@{1+a-d}\EulerGamma@{1+a-b-c-d}} {\EulerGamma@{1+a}\EulerGamma@{1+a-b-c}\EulerGamma@{1+a-b-d}\EulerGamma@{1+a-c-d}} } \HyperpFq 43 @ @ 1 + a / 2 , a , b , c a / 2 , 1 + a - b , 1 + a - c - 1 = Γ ( 1 + a - b ) Γ ( 1 + a - c ) Γ ( 1 + a ) Γ ( 1 + a - b - c ) \HyperpFq 43 @ @ 1 𝑎 2 𝑎 𝑏 𝑐 𝑎 2 1 𝑎 𝑏 1 𝑎 𝑐 1 Euler-Gamma 1 𝑎 𝑏 Euler-Gamma 1 𝑎 𝑐 Euler-Gamma 1 𝑎 Euler-Gamma 1 𝑎 𝑏 𝑐 {\displaystyle{\displaystyle{\displaystyle\HyperpFq{4}{3}@@{1+a/2,a,b,c}{a/2,1% +a-b,1+a-c}{-1}=\frac{\Gamma\left(1+a-b\right)\Gamma\left(1+a-c\right)}{\Gamma% \left(1+a\right)\Gamma\left(1+a-b-c\right)}}}} {\displaystyle \HyperpFq{4}{3}@@{1+a/2,a,b,c}{a/2,1+a-b,1+a-c}{-1} =\frac{\EulerGamma@{1+a-b}\EulerGamma@{1+a-c}}{\EulerGamma@{1+a}\EulerGamma@{1+a-b-c}} } ∑ n = - ∞ ∞ Γ ( n + a ) Γ ( n + b ) Γ ( n + c ) Γ ( n + d ) = Γ ( a ) Γ ( 1 - a ) Γ ( b ) Γ ( 1 - b ) Γ ( c + d - a - b - 1 ) Γ ( c - a ) Γ ( d - a ) Γ ( c - b ) Γ ( d - b ) superscript subscript 𝑛 Euler-Gamma 𝑛 𝑎 Euler-Gamma 𝑛 𝑏 Euler-Gamma 𝑛 𝑐 Euler-Gamma 𝑛 𝑑 Euler-Gamma 𝑎 Euler-Gamma 1 𝑎 Euler-Gamma 𝑏 Euler-Gamma 1 𝑏 Euler-Gamma 𝑐 𝑑 𝑎 𝑏 1 Euler-Gamma 𝑐 𝑎 Euler-Gamma 𝑑 𝑎 Euler-Gamma 𝑐 𝑏 Euler-Gamma 𝑑 𝑏 {\displaystyle{\displaystyle{\displaystyle{}{}\sum_{n=-\infty}^{\infty}\frac{% \Gamma\left(n+a\right)\Gamma\left(n+b\right)}{\Gamma\left(n+c\right)\Gamma% \left(n+d\right)}=\frac{\Gamma\left(a\right)\Gamma\left(1-a\right)\Gamma\left(% b\right)\Gamma\left(1-b\right)\Gamma\left(c+d-a-b-1\right)}{\Gamma\left(c-a% \right)\Gamma\left(d-a\right)\Gamma\left(c-b\right)\Gamma\left(d-b\right)}}}} {\displaystyle \index{Dougall's bilateral sum}\index{Summation formula!Dougall} \sum_{n=-\infty}^{\infty}\frac{\EulerGamma@{n+a}\EulerGamma@{n+b}}{\EulerGamma@{n+c}\EulerGamma@{n+d}} =\frac{\EulerGamma@{a}\EulerGamma@{1-a}\EulerGamma@{b}\EulerGamma@{1-b}\EulerGamma@{c+d-a-b-1}} {\EulerGamma@{c-a}\EulerGamma@{d-a}\EulerGamma@{c-b}\EulerGamma@{d-b}} } Γ ( n + a ) Γ ( n + b ) = Γ ( a ) Γ ( 1 - a ) Γ ( b ) Γ ( 1 - b ) Γ ( 1 - a - n ) Γ ( 1 - b - n ) Euler-Gamma 𝑛 𝑎 Euler-Gamma 𝑛 𝑏 Euler-Gamma 𝑎 Euler-Gamma 1 𝑎 Euler-Gamma 𝑏 Euler-Gamma 1 𝑏 Euler-Gamma 1 𝑎 𝑛 Euler-Gamma 1 𝑏 𝑛 {\displaystyle{\displaystyle{\displaystyle\Gamma\left(n+a\right)\Gamma\left(n+% b\right)=\frac{\Gamma\left(a\right)\Gamma\left(1-a\right)\Gamma\left(b\right)% \Gamma\left(1-b\right)}{\Gamma\left(1-a-n\right)\Gamma\left(1-b-n\right)}}}} {\displaystyle \EulerGamma@{n+a}\EulerGamma@{n+b}=\frac{\EulerGamma@{a}\EulerGamma@{1-a}\EulerGamma@{b}\EulerGamma@{1-b}} {\EulerGamma@{1-a-n}\EulerGamma@{1-b-n}} }
∑ n = - ∞ ∞ 1 Γ ( n + c ) Γ ( n + d ) Γ ( 1 - a - n ) Γ ( 1 - b - n ) = Γ ( c + d - a - b - 1 ) Γ ( c - a ) Γ ( d - a ) Γ ( c - b ) Γ ( d - b ) superscript subscript 𝑛 1 Euler-Gamma 𝑛 𝑐 Euler-Gamma 𝑛 𝑑 Euler-Gamma 1 𝑎 𝑛 Euler-Gamma 1 𝑏 𝑛 Euler-Gamma 𝑐 𝑑 𝑎 𝑏 1 Euler-Gamma 𝑐 𝑎 Euler-Gamma 𝑑 𝑎 Euler-Gamma 𝑐 𝑏 Euler-Gamma 𝑑 𝑏 {\displaystyle{\displaystyle{\displaystyle\sum_{n=-\infty}^{\infty}\frac{1}{% \Gamma\left(n+c\right)\Gamma\left(n+d\right)\Gamma\left(1-a-n\right)\Gamma% \left(1-b-n\right)}{}=\frac{\Gamma\left(c+d-a-b-1\right)}{\Gamma\left(c-a% \right)\Gamma\left(d-a\right)\Gamma\left(c-b\right)\Gamma\left(d-b\right)}}}} {\displaystyle \sum_{n=-\infty}^{\infty}\frac{1}{\EulerGamma@{n+c}\EulerGamma@{n+d}\EulerGamma@{1-a-n}\EulerGamma@{1-b-n}} {}=\frac{\EulerGamma@{c+d-a-b-1}}{\EulerGamma@{c-a}\EulerGamma@{d-a}\EulerGamma@{c-b}\EulerGamma@{d-b}} }