Abstract
A variety of identities involving harmonic numbers and generalized harmonic numbers have been investigated since the distant past and involved in a wide range of diverse fields such as analysis of algorithms in computer science, various branches of number theory, elementary particle physics and theoretical physics. Here we show how one can obtain further interesting identities about certain finite series involving binomial coefficients, harmonic numbers and generalized harmonic numbers by applying the usual differential operator to a known identity. MSC:11M06, 33B15, 33E20, 11M35, 11M41, 40C15.
Highlights
Introduction and preliminariesThe generalized harmonic numbers Hn(s) of order s are defined by Hn(s) :=n js (n ∈ N; s ∈ C), ( . ) j= and Hn := Hn( ) = j j= (n ∈ N)are the harmonic numbers
We aim at presenting further interesting identities about certain interesting finite series associated with binomial coefficients, harmonic numbers and generalized harmonic numbers
We will try to express a class of the following finite sums involving harmonic numbers and binomial coefficients as given above: n (– )j+ (j + )k n j Hj (n ∈ N ; k ∈ N)
Summary
Spivey [ ] presented many summation formulas involving binomial coefficients, the Stirling numbers of the first and second kind and harmonic numbers, two of which are chosen to be recalled here: [ , Identity ] Paule and Schneider [ ] proved five conjectured harmonic number identities similar to those arising in the context of supercongruences for Apéry numbers, one of which is recalled here as follows [ , Eq ( )]: n ] recorded six commonly used identities that involve both binomial coefficients and harmonic numbers, two of which are recalled here: n
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