Further Ramanujan-like series containing harmonic numbers and squared binomial coefficients (Q780450)

Using the hypergeometric series method the authors present a systematic evaluation of series of the form \(\sum _{n=1}^{\infty }{\binom {2n} {n}}^{2} \frac{H_{n} }{16^{n} P(n)} ,\; \; \sum _{n=1}^{\infty }{\binom {2n} {n}}^{2} \frac{H_{n} }{16^{n} Q(n)} ,\; \sum _{n=1}^{\infty }{\binom {2n} {n}}^{2} \frac{H_{n}^{(2)} }{16^{n} P(n)} \), and \(\sum _{n=1}^{\infty }{\binom {2n} {n}}^{2} \frac{H_{n}^{(2)} }{16^{n} Q(n)} \). Here \(P(n)\) are certain linear polynomials and \(Q(n)\) are certain linear or quadratic polynomials; \(H_{n} \) are the harmonic numbers and \(H_{n}^{(2)} =1+2^{-2} +3^{-2} +...+n^{-2} \).