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

2010/05/16 by Victor J. W. Guo, Guo, Victor J. W.
Computer Science · Mathematics · #05A10 #05A19 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05A10 #msc:05A19 #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1005.2745

9 pages

arxiv created 2010/05/16 · openalex publication_date 2010/05/16 · arxiv updated 2010/05/18 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

Motivated by the recent work of Chu [Electron. J. Combin. 17 (2010), #N24], we give simple proofs of Jensen's identity ∑_k=0ⁿx+kz\choose ky-kz\choose n-k =∑_k=0ⁿx+y-k\choose n-kz^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 ∑_{k≥ 0}m+r\choose m-n-kn+k\choose nx^m-n-ky^k =∑_{k≥ 0}-r\choose m-n-kn+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.

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