Rothe–Hagen identity

Short description: Generalization of Vandermonde's identity

In mathematics, the Rothe–Hagen identity is a mathematical identity valid for all complex numbers ([math]\displaystyle{ x, y, z }[/math]) except where its denominators vanish:

[math]\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}. }[/math]

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, doi:10.37236/473, http://www.combinatorics.org/ojs/index.php/eljc/article/view/v17i1n24 .
• Gould, H. W. (1956), "Some generalizations of Vandermonde's convolution", The American Mathematical Monthly 63 (2): 84–91, doi:10.1080/00029890.1956.11988763 . See especially pp. 89–91.
• Synopsis Der Hoeheren Mathematik, Berlin, 1891, 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 .
• Rothe, Heinrich August (1793), Formulae De Serierum Reversione Demonstratio Universalis Signis Localibus Combinatorio-Analyticorum Vicariis Exhibita: Dissertatio Academica, Leipzig . As cited by (Gould 1956).