Ramanujan-like series for $\frac{1}{\pi }$ involving harmonic numbers

Abstract

We introduce new classes of Ramanujan-like series for $\frac{1}{\pi }$, by devising methods for evaluating harmonic sums involving squared central binomial coefficients, such as the Ramanujan-type series

$$\begin{aligned} \sum _{n=1}^{\infty } \frac{\left( {\begin{array}{c}2 n\\ n\end{array}}\right) ^2 \left( H_n^2+H_n^{(2)}\right) }{16^n (2 n-1)} = \frac{4 \pi }{3}-\frac{32 \ln ^2(2) - 32 \ln (2) + 16 }{\pi } \end{aligned}$$

introduced in this article. While the main technique used in this article is based on the evaluation of a parameter derivative of a beta-type integral, we also show how new integration results involving complete elliptic integrals may be used to evaluate Ramanujan-like series for $\frac{1}{\pi }$ containing harmonic numbers.