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I'm looking for books about harmonic numbers, where I could find proofs of results about them. For example a proof for the fact, that the generating function of the generalized harmonic numbers is $$ \sum_{n=1}^\infty H_{n}^{(m)} z^n = \frac {\mathrm{Li}_m(z)}{1-z}. $$ I'm also interested in particular evaluations of sums involving harmonic numbers, like this: $$ \sum_{n=1}^{\infty}\frac{H_n}{n^2 2^n}=\zeta(3)-\frac{1}{2}\ln(2)\zeta(2). $$ I'm also interested in sums of generalized harmonic numbers and also alternating harmonic numbers.

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  • $\begingroup$ Have a look at this - Not really a book, but a MathWorld@Wolfram article on harmonic numbers. $\endgroup$ – Jose Arnaldo Bebita-Dris Oct 1 '15 at 16:53
  • $\begingroup$ @JoseArnaldoBebitaDris Thank you, I've seen this, but the article is not so good. There are a lot of evaluation marked as "pers. comm.", and there is no reference of the proofs. $\endgroup$ – user153012 Oct 1 '15 at 17:26
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Zeta and q-Zeta Functions and Associated Series and Integrals (Elsevier Insights) and the papers referenced therein regarding harmonic numbers and generalised harmonic numbers has a fair bit of info.

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You might be interested in Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values, by Jianqiang Zhao.

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For a multitude of problems on infinite sums involving the harmonic and generalised harmonic numbers, so-called Euler sums, you cannot go pass Cornel Ioan Valean's latest book: (Almost) Impossible Integrals, Sums, and Series.

Solutions to all problems are given and some of the sums are very challenging indeed. It is an absolute must have book if you are seriously interested in such things.

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