Ramanujan-like series for $\frac{1}{\pi}$ involving harmonic numbers
Résumé
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 $$\sum_{n=1}^{\infty} \frac{\binom{2 n}{n}^2 \left(H_n^2+H_n^{(2)}\right)}{16^n (2 n-1)} = \frac{4 \pi}{3}-\frac{32 \ln^2(2) - 32 \ln (2) + 16 }{\pi}$$ 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.
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