Parametric binomial sums involving harmonic numbers

We present explicit formulas for the following family of parametric binomial sums involving harmonic numbers for p = 0, 1,2 and vertical bar t vertical bar <= 1. Sigma(infinity)(k=1) H(k-1)t(k)/k(p)(n+k(k)) and Sigma(infinity)(k=1) t(k)/k(p)(n+k(k)). We also generalize the following relation between the Stirling numbers of the first kind and the Riemann zeta function to polygamma function and give some applications. zeta(n + 1) = Sigma(infinity)(k=n) s(k,n)/kk!, n = 1, 2, 3, .... As examples, zeta(3) = 1/7 Sigma(infinity)(k=1)H(k-1)4(k)/k(2)(2k(k)), and zeta(3) = 8/7 + 1/7 Sigma(infinity)(k=1) H(k-1)4(k)/k(2)(2k + 1)(2k(k)), which are new series representations for the Apery constant zeta(3).

Automated summary

The paper presents explicit formulas for a family of parametric binomial sums involving harmonic numbers for various values of p and |t| <= 1. It also generalizes a relation between Stirling numbers and the Riemann zeta function to polygamma function and provides some new series representations for the Apery constant zeta(3).