Abel's binomial theorem

{{short description|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:

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

Example

=The case ''m'' = 2=

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

References

{{mathworld|title=Abel's binomial theorem|urlname=AbelsBinomialTheorem}}

Category:Factorial and binomial topics

Category:Theorems in algebra

Abel's binomial theorem

Example

=The case ''m'' = 2=

See also

References