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Motivated by the recent work of Chu [Electron. J. Combin. 17 (2010), #N24],we give simple proofs of Jensen's identity $$ \sum_{k=0}^{n}{x+kz\choosek}{y-kz\choose n-k} =\sum_{k=0}^{n}{x+y-k\choose n-k}z^k, $$ and Chu's andMohanty-Handa's generalizations of Jensen's identity. We also give a quitesimple proof of an equivalent form of Graham-Knuth-Patashnik's identity $$\sum_{k\geq 0}{m+r\choose m-n-k}{n+k\choose n}x^{m-n-k}y^k =\sum_{k\geq0}{-r\choose m-n-k}{n+k\choose n}(-x)^{m-n-k}(x+y)^k, $$ which wasrediscovered, respectively, by Sun in 2003 and Munarini in 2005. Finally wegive a multinomial coefficient generalization of this identity and raise twoopen problems.

Simple proofs of Jensen's, Chu's, Mohanty-Handa's, and Graham-Knuth-Patashnik's identities