On some combinatorial identities and harmonic sums

Abstract

For any $m,n\in \mathbb{N}$ we first give new proofs for the following well known combinatorial identities \begin{equation*} S_n(m)=\sum\limits_{k=1}^n\binom{n}{k}\frac{(-1)^{k-1}}{k^m}=\sum\limits_{n\geq r_1\geq r_2\geq...\geq r_m\geq 1}\frac{1}{r_1r_2\cdots r_m} \end{equation*} and $$\sum _{k=1}^n(-1{)}^{n-k}(\genfrac{}{}{0ex}{}{n}{k})k^n=n!,$$ and then we produce the generating function and an integral representation for $S_n(m)$. Using them we evaluate many interesting finite and infinite harmonic sums in closed form. For example, we show that $$\zeta (3)=\frac{1}{9}\sum _{n=1}^{\infty }\frac{H_n^3+3H_n{H}_n^{(2)}+2H_n^{(3)}}{2^n},$$ and $$\zeta (5)=\frac{2}{45}\sum _{n=1}^{\infty }\frac{H_n^4+6H_n^2{H}_n^{(2)}+8H_n{H}_n^{(3)}+3{\left(H_n^{(2)}\right)}^2+6H_n^{(4)}}{n{2}^n},$$ where $H_n^{(i)}$ are generalized harmonic numbers defined below.

to appear in Int. J. Number Theory, 2017

References in corpus (1)

• Abel's Lemma and Identities on Harmonic Numbers