Generalization of Vandermonde's identity
In mathematics , the Rothe–Hagen identity is a mathematical identity valid for all complex numbers (
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{\displaystyle x,y,z}
vanish :
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.
{\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, doi :10.37236/473 Gould, H. W. (1956), "Some generalizations of Vandermonde's convolution", The American Mathematical Monthly 63 (2): 84– 91, doi :10.1080/00029890.1956.11988763 , JSTOR 2306429 , MR 0075170 Hagen, Johann G. (1891), Synopsis Der Hoeheren Mathematik , Berlin, formula 17, pp. 64–68, vol. I{{citation }}: CS1 maint: location missing publisher (link )Gould (1956) .Ma, Xinrong (2011), "Two matrix inversions associated with the Hagen-Rothe formula, their q -analogues and applications", Journal of Combinatorial Theory 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 Gould (1956) .
