Rothe–Hagen identity

In mathematics, the Rothe–Hagen identity is a mathematical identity valid for all complex numbers ( $x, y, z$ ) except where the denominators vanish:

\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.

References

Chu, Wenchang (2010), "Elementary proofs for convolution identities of Abel and Hagen-Rothe", Electronic Journal of Combinatorics 17 (1), N24 .
Gould, H. W. (1956), "Some generalizations of Vandermonde's convolution", The American Mathematical Monthly 63: 84–91, JSTOR 2306429, MR 0075170 . See especially pp. 89–91.
Hagen, Johann G. (1891), Synopsis Der Hoeheren Mathematik, Berlin, formula 17, pp. 64–68, vol. I . As cited by Gould (1956).
Ma, Xinrong (2011), "Two matrix inversions associated with the Hagen-Rothe formula, their q-analogues and applications", Journal of Combinatorial Theory, Series A 118 (4): 1475–1493, doi:10.1016/j.jcta.2010.12.012, MR 2763069 .
Rothe, Heinrich August (1793), Formulae De Serierum Reversione Demonstratio Universalis Signis Localibus Combinatorio-Analyticorum Vicariis Exhibita: Dissertatio Academica, Leipzig . As cited by Gould (1956).