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:

$\sum _{{k=0}}^{m}{\binom {m}{k}}(w+m-k)^{{m-k-1}}(z+k)^{k}=w^{{-1}}(z+w+m)^{m}.$

Example

m = 2

${\begin{aligned}&{}\quad {\binom {2}{0}}(w+2)^{1}(z+0)^{0}+{\binom {2}{1}}(w+1)^{0}(z+1)^{1}+{\binom {2}{2}}(w+0)^{{-1}}(z+2)^{2}\\&=(w+2)+2(z+1)+{\frac {(z+2)^{2}}{w}}\\&={\frac {(z+w+2)^{2}}{w}}.\end{aligned}}$

See also

• Binomial theorem
• Binomial type

References

• Weisstein, Eric W. "Abel's binomial theorem". MathWorld.