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:

${\displaystyle \sum _{k=0}^{n}{\frac {x}{x+kz}}{x+kz \choose k}{\frac {y}{y+(n-k)z}}{y+(n-k)z \choose n-k}={\frac {x+y}{x+y+nz}}{x+y+nz \choose n}.}$

It is a generalization of Vandermonde's identity, and is named after Heinrich August Rothe and Johann Georg Hagen.

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).