Series Containing Squared Central Binomial Coefficients and Alternating Harmonic Numbers

Abstract

We present a new integration method for evaluating infinite series involving alternating harmonic numbers. Using this technique, we provide new evaluations for series containing factors of the form $\left( {\begin{array}{c}2n\\ n\end{array}}\right) ^{2} H_{2n}'$ that cannot be evaluated using known generating functions involving harmonic-type numbers. A closed-form evaluation is given for the series:

$$\begin{aligned} \sum _{n = 1}^{\infty } \left( - \frac{1}{16} \right) ^{n} \frac{ \left( {\begin{array}{c}2 n\\ n\end{array}}\right) ^2 H_{2n}' }{ n + 1}, \end{aligned}$$

and we show how the integration method given in our article may be applied to evaluate natural generalizations and variants of the above series, such as the binomial-harmonic series:

$$\begin{aligned} \sum _{n = 1}^{\infty } \frac{\left( -\frac{1}{16}\right) ^n \left( {\begin{array}{c}2 n\\ n\end{array}}\right) ^2 H_{2n}' }{2 n - 1} = \frac{(\pi -4 \ln (2)) \Gamma ^2\left( \frac{1}{4}\right) }{8 \sqrt{2} \pi ^{3/2}} - \frac{\sqrt{\frac{\pi }{2}} (\pi +4 \ln (2) - 4 )}{\Gamma ^2\left( \frac{1}{4}\right) } \end{aligned}$$

introduced in our article.