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<p><b>New page</b></p><div>In [[mathematics]], the '''Rothe–Hagen identity''' is a [[mathematical identity]] valid for all [[complex number]]s (<math>x, y, z</math>) except where the denominators vanish:<br />
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:<math>\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><br />
<br />
It is a generalization of [[Vandermonde's identity]], and is named after [[Heinrich August Rothe]] and [[Johann Georg Hagen]].<br />
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==References==<br />
*{{citation<br />
| last = Chu | first = Wenchang<br />
| issue = 1<br />
| journal = [[Electronic Journal of Combinatorics]]<br />
| at = N24<br />
| title = Elementary proofs for convolution identities of Abel and Hagen-Rothe<br />
| url = http://www.combinatorics.org/ojs/index.php/eljc/article/view/v17i1n24<br />
| volume = 17<br />
| year = 2010}}.<br />
*{{citation<br />
| last = Gould | first = H. W.<br />
| journal = [[The American Mathematical Monthly]]<br />
| jstor = 2306429<br />
| mr = 0075170<br />
| pages = 84–91<br />
| title = Some generalizations of Vandermonde's convolution<br />
| volume = 63<br />
| year = 1956}}. See especially pp.&nbsp;89–91.<br />
*{{citation<br />
| last = Hagen | first = Johann G. | authorlink = Johann Georg Hagen<br />
| title = Synopsis Der Hoeheren Mathematik<br />
| at = formula 17, pp. 64–68, vol. I<br />
| location = Berlin<br />
| year = 1891}}. As cited by {{harvtxt|Gould|1956}}.<br />
*{{citation<br />
| last = Ma | first = Xinrong<br />
| doi = 10.1016/j.jcta.2010.12.012<br />
| issue = 4<br />
| journal = [[Journal of Combinatorial Theory]] | series = Series A<br />
| mr = 2763069<br />
| pages = 1475–1493<br />
| title = Two matrix inversions associated with the Hagen-Rothe formula, their ''q''-analogues and applications<br />
| volume = 118<br />
| year = 2011}}.<br />
*{{citation<br />
| last = Rothe | first = Heinrich August<br />
| title = Formulae De Serierum Reversione Demonstratio Universalis Signis Localibus Combinatorio-Analyticorum Vicariis Exhibita: Dissertatio Academica<br />
| url = http://books.google.com/books/about/Formulae_De_Serierum_Reversione_Demonstr.html<br />
| location = Leipzig<br />
| year = 1793}}. As cited by {{harvtxt|Gould|1956}}.<br />
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{{DEFAULTSORT:Rothe-Hagen identity}}<br />
[[Category:Factorial and binomial topics]]<br />
[[Category:Mathematical identities]]<br />
[[Category:Complex analysis]]<br />
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{{mathapplied-stub}}</div>en>Qetuth