# Abel's binomial theorem

Abel's binomial theorem 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}.} Contents 1 Example 1.1 The case m = 2 2 See also 3 References Example The case m = 2 {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}}} See also Binomial theorem Binomial type References Weisstein, Eric W. "Abel's binomial theorem". MathWorld.

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