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Mathematical identity involving sums of binomial coefficients
Abel's binomial theorem, named after Niels Henrik Abel, is a mathematical identity involving sums of binomial coefficients. It states the following:
![{\displaystyle \sum _{k=0}^{m}{\binom {m}{k}}(w+m-k)^{m-k-1}(z+k)^{k}=w^{-1}(z+w+m)^{m}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/825e00d7429fb08707af57e48cc9d41eaa28e0b2)
Example[edit]
The case m = 2[edit]
![{\displaystyle {\begin{aligned}&{}\quad {\binom {2}{0}}(w+2)^{1}(z+0)^{0}+{\binom {2}{1}}(w+1)^{0}(z+1)^{1}+{\binom {2}{2}}(w+0)^{-1}(z+2)^{2}\\&=(w+2)+2(z+1)+{\frac {(z+2)^{2}}{w}}\\&={\frac {(z+w+2)^{2}}{w}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f794b34f45df24a6f9d5312a96d3e5392cddd1a)
See also[edit]
References[edit]