Abstract
Three new closed-form summation formulae involving harmonic numbers are established using simple arguments and they are very general extensions of Euler's famous harmonic sum identity. Some illustrative special cases as well as immediate consequences of the main results are also considered.
Highlights
About 235 years ago, circa 1775, Euler [7], produced one of his many remarkable identities, namely (1) ∞ Hn nq = (q + 2) ζ + 1) −q−2 ζ (m + 1) ζ m) n =1 m =1(q ∈ N \ {1}, N := {1, 2, 3, . . .}), n where Hn := m−1, n ∈ N, is the nth harmonic number, while ζ(z) denotes the m=1 familiar Riemann Zeta function
That, what remains is to sum Hn (n(an + r))−1 and Hnn−s. It is clear n=1 n=1 that the latter series can be summed by using the Euler result (1), while for the former, by Lemma 1, we deduce the following summation formula
In the beginning of this section, for the sake of completeness of the proof of Lemma 1, we prove the summation formula given by (7) and deduce the following integral formula
Summary
We define and use the x−generalized harmonic number in power r, Hx(r), given in terms of the polygamma functions as The four out of six series obtained in this way are well-known and very simple to sum It is easy to find their summation formulae in any standard reference book (see, for instance, [8, Section 5.1]) and to sum them it suffices to make a partial fraction decomposition and recall that ψ(z+1) = −γ+ z(n(n+z))−1, ψ′(z+1) =
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