Sums involving the binomial coefficients, Bernoulli numbers of the second kind and harmonic numbers

Abstract

We offer a number of various finite and infinite sum identities involving the binomial coefficients, Bernoulli numbers of the second kind and harmonic numbers. For example, among many others, we prove \[\displaystyle \sum_{k=0}^{n}\frac{(-1)^{k}h_{k}}{4^{k}} {{2k} \choose {k}}G_{n-k}=\frac{(-1)^{n-1}}{2^{2n-1}}{{2n-2} \choose {n-1}}\] and \[\displaystyle \sum_{k=1}^{\infty}\frac{h_{k}}{k^{2}(2k-1)4^{k}} {{2k} \choose {k}}=2\pi +3\zeta(2)\log 2-3\zeta(2)-\frac{7}{2}\zeta(3),\] where h_k=H_{2k}-\dfrac{1}{2}H_{k}, G_k are Bernoulli numbers of the second kind, and \zeta is the Riemann zeta function. We also give an alternate proof of the series representations for the constants \log (2 \pi) and \gamma given by Blagouchine and Coppo.