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

TL;DR

This paper provides simplified proofs of several classical binomial and multinomial identities, including Jensen's, Chu's, Mohanty-Handa's, and Graham-Knuth-Patashnik's, along with generalizations and open problems.

Contribution

It introduces straightforward proofs for multiple well-known identities and their generalizations, making these results more accessible and extending them to multinomial coefficients.

Findings

Simplified proofs of Jensen's, Chu's, Mohanty-Handa's identities

A simple proof of Graham-Knuth-Patashnik's identity

A multinomial coefficient generalization and open problems

Abstract

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