On Jensen's and related combinatorial identities

Abstract

Motivated by the recent work of Chu [Electron. J. Combin. 17 (2010), #N24], we give simple proofs of Jensen's identity n?k=0 (x+kz/k)(y-kz/n-k = n?k=0 (x+y-k)/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 ?k?0 (m+r/m-n-k) (n+k/n)xm-n-k yk = ?k?0 (-r/m-n-k)(n+k/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.