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Simple proofs of Jensen's, Chu's, Mohanty-Handa's, and Graham-Knuth-Patashnik's identities (1005.2745v1)
Published 16 May 2010 in math.CO
Abstract: Motivated by the recent work of Chu [Electron. J. Combin. 17 (2010), #N24], we give simple proofs of Jensen's identity $$ \sum_{k=0}{n}{x+kz\choose k}{y-kz\choose n-k} =\sum_{k=0}{n}{x+y-k\choose n-k}zk, $$ and Chu's and Mohanty-Handa's generalizations of Jensen's identity. We also give a quite simple proof of an equivalent form of Graham-Knuth-Patashnik's identity $$ \sum_{k\geq 0}{m+r\choose m-n-k}{n+k\choose n}x{m-n-k}yk =\sum_{k\geq 0}{-r\choose m-n-k}{n+k\choose n}(-x){m-n-k}(x+y)k, $$ which was rediscovered, respectively, by Sun in 2003 and Munarini in 2005. Finally we give a multinomial coefficient generalization of this identity and raise two open problems.