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 and (almost) serendipitous identities about certain finite or infinite series involving binomial coefficients, harmonic numbers, and generalized harmonic numbers by simply applying the usual differential operator to well-known Gauss’s summation formula for2F1(1).
Highlights
Introduction and PreliminariesThe generalized harmonic numbers Hn(s) of order s which are defined by Hn(s) := n ∑ j=11 js (n ∈ N; s ∈ C), (1) Hn Hn(1) =
Setting c = 2 and a = b = 1 in (31) to (34) and using some suitable identities in Section 1, we obtain some interesting identities involving harmonic numbers and generalized harmonic numbers given in the following corollary
Differentiating each side of (28) with respect to the variable c successively and using some suitable identities in Section 1 and Lemma 1, we obtain a set of infinite series involving binomial coefficients, harmonic numbers, and generalized harmonic numbers different from those in Theorem 2 as in the following theorem
Summary
Department of Mathematics, Dongguk University, Gyeongju 780-714, Republic of Korea 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 and (almost) serendipitous identities about certain finite or infinite series involving binomial coefficients, harmonic numbers, and generalized harmonic numbers by simply applying the usual differential operator to well-known Gauss’s summation formula for 2F1(1).
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