Abel's binomial theorem

Abel's binomial theorem, named after Niels Henrik Abel, states the following:

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

Example

m=2

${\binom {2}{0}}(w+2)^{1}(z+0)^{0}+{\binom {2}{1}}(w+1)^{0}(z+1)^{1}+{\binom {2}{2}}(w+0)^{-1}(z+2)^{2}$

$=(w+2)+2(z+1)+{\frac {(z+2)^{2}}{w}}$

$={\frac {(z+w+2)^{2}}{w}}.$

See also

• Binomial theorem
• Binomial type

References

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