Series containing squared central binomial coefficients and alternating harmonic numbers
Résumé
We introduce an integration method for evaluating infinite series involving alternating harmonic numbers that generalizes many of the results in [Integral Transforms Spec Funct 28:7, 2017]. Using this method, we provide new evaluations for series containing factors of the form $\binom{2n}{n}^{2} H_{2n}'$ that cannot be evaluated using known generating functions involving harmonic-type numbers. A closed-form evaluation is given for the series $$\sum _{n = 1}^{\infty } \left( - \frac{1}{16} \right)^{n} \frac{ \binom{2 n}{n}^2 H_{2n}' }{ n + 1},$$ and we describe why the problem of symbolically computing this particular series is especially difficult. The integration technique given in our article may be applied to evaluate natural generalizations and variants of the above series, such as the binomial-harmonic series $$\sum _{n = 1}^{\infty } \frac{\left(-\frac{1}{16}\right)^n
\binom{2 n}{n}^2 H_{2n}' }{2 n - 1} = \frac{(\pi -4 \ln (2)) \Gamma^2\big(\frac{1}{4}\big)}{8 \sqrt{2} \pi ^{3/2}} - \frac{\sqrt{\frac{\pi }{2}} (\pi +4 \ln (2) - 4 )}{\Gamma^2\big(\frac{1}{4}\big)}$$ introduced in our article. Our integration-based method, when applied in conjunction with the main lemma from [Ramanujan J 46:2, 2018], also provides a new way of evaluating series involving even-indexed harmonic numbers.
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