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

Abstract

Motivated by the recent work of Chu [Electron. J. Combin. 17 (2010), #N24], we give simple proofs of Jensen's identity $$\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{x+kz}{k}\right)\left(\genfrac{}{}{0ex}{}{y-kz}{n-k}\right)=\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{x+y-k}{n-k}\right)z^k,$$ 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 $$\sum _{k\ge 0}\left(\genfrac{}{}{0ex}{}{m+r}{m-n-k}\right)\left(\genfrac{}{}{0ex}{}{n+k}{n}\right)x^{m-n-k}{y}^k=\sum _{k\ge 0}\left(\genfrac{}{}{0ex}{}{-r}{m-n-k}\right)\left(\genfrac{}{}{0ex}{}{n+k}{n}\right)(-x{)}^{m-n-k}(x+y{)}^k,$$ 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.

arXiv comment9 pages

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