Abel's binomial theorem

Template:Short description 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

\begin{aligned} (\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}