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.

• Have a look at this - Not really a book, but a MathWorld@Wolfram article on harmonic numbers. – Arnie Bebita-Dris Oct 1 '15 at 16:53
• @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. – user153012 Oct 1 '15 at 17:26