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:

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

Example

The case m = 2

$\begin{array}{rl}& (\genfrac{}{}{0ex}{}{2}{0})(w+2{)}^1(z+0{)}^0+(\genfrac{}{}{0ex}{}{2}{1})(w+1{)}^0(z+1{)}^1+(\genfrac{}{}{0ex}{}{2}{2})(w+0{)}^{-1}(z+2{)}^2\\ & =(w+2)+2(z+1)+\frac{(z+2{)}^2}{w}\\ & =\frac{(z+w+2{)}^2}{w}.\end{array}$

See also

• Binomial theorem
• Binomial type

References

• Weisstein, Eric W.. "Abel's binomial theorem". http://mathworld.wolfram.com/AbelsBinomialTheorem.html.