Harmonic sums from the Kummer theorem (Q2033210)

In this article, several results on the hypergeometric series due to Euler, Gauss, Kummer and Ramanujan are generalized. The main results are closed evaluations of series like \[ \sum_{n=0}^\infty \Big(- \frac1{16}\Big)^n \frac{2n}{n}^2 H_n(\frac{1}{2}-\lambda)(\frac{1}{2}-n)_\lambda \] in terms of the hypergeometric function, where \(H_n(x) = \frac{1}{x} + \frac{1}{x+1} + \ldots + \frac{1}{x+n-1}\) and \((x)_n = x(x+1) \cdots (x+n-1)\).