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

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

See also

• Binomial theorem
• Binomial type

References

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