Formula:KLS:01.05:02

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\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}}}}

Constraint(s)

n = 0 , 1 , 2 , 𝑛 0 1 2 {\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots}}}


Proof

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Symbols List

F q p subscript subscript 𝐹 𝑞 𝑝 {\displaystyle{\displaystyle{\displaystyle{{}_{p}F_{q}}}}}  : generalized hypergeometric function : http://dlmf.nist.gov/16.2#E1
Σ Σ {\displaystyle{\displaystyle{\displaystyle\Sigma}}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
( a ) n subscript 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
( n k ) binomial 𝑛 𝑘 {\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}  : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1

Bibliography

Equation in Section 1.5 of KLS.

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