Combinatorial identities involving harmonic numbers

2018/06/08 by Batir, Necdet
#05A10 #05A19 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1806.03022

Abstract

In this work we prove a new combinatorial identity and applying it we establish many finite harmonic sum identities. Among many others, we prove that ∑_k=1ⁿ\frac(-1)^k-1k\binomnkH_n-k=H_n²+∑_k=1ⁿ\frac(-1)^kk²\binomnk, and ∑_k=1ⁿ\frac(-1)^k-1k²\binomnkH_n-k=\fracH_n[H_n²+H_n⁽²⁾]2-∑_k=0^n-1\frac(-1)^k[H_n-H_k](k+1)(n-k)\binomnk. Almost all of our results are new, while a few of them recapture know results.

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