Rothe–Hagen identity
In mathematics, the Rothe–Hagen identity is a mathematical identity valid for all complex numbers ( x , y , z {\displaystyle x,y,z} ) except where its denominators vanish: ∑ k = 0 n x x + k z ( x + k z k ) y y + ( n − k ) z ( y + ( n − k ) z n − k ) = x + y x + y + n z ( x + y + n z n ) . {\displaystyle \sum _{k=0}^{n}{\frac {x}{x+kz}}{x+kz \choose k}{\frac {y}{y+(n-k)z}}{y+(n-k)z \choose n-k}={\frac {x+y}{x+y+nz}}{x+y+nz \choose n}.} It is a generalization of Vandermonde's identity, and is named after Heinrich August Rothe and Johann Georg Hagen.