# Series Containing Squared Central Binomial Coefficients and Alternating Harmonic Numbers

• John M. Campbell
Article

## 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.

## Keywords

Alternating harmonic number infinite series integral transform harmonic series

## Mathematics Subject Classification

Primary 33C75 33C20 Secondary 65B10

## References

1. 1.
Boyadzhiev, K.N.: Series with central binomial coefficients, Catalan numbers, and harmonic numbers. J. Integer Seq. 15(1), Article 12.1.7, 11 (2012)
2. 2.
Campbell, J.M.: New series involving harmonic numbers and squared central binomial coefficients. HAL e-prints. https://hal.archives-ouvertes.fr/hal-01774708 (2018)
3. 3.
Campbell, J.M.: Ramanujan-like series for $$\frac{1}{\pi }$$ involving harmonic numbers. Ramanujan J. 46, 373–387 (2018)
4. 4.
Campbell, J.M., Sofo, A.: An integral transform related to series involving alternating harmonic numbers. Integral Transforms Spec. Funct. 28, 547–559 (2017)
5. 5.
Campbell, J.M., D’Aurizio, J., Sondow, J.: On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier–Legendre series expansions. ArXiv e-prints. arXiv:1710.03221v2 (2018)
6. 6.
Cantarini, M., D’Aurizio, J.: On the interplay between hypergeometric series, Fourier–Legendre expansions and Euler sums. ArXiv e-prints. arXiv:1806.08411 (2018)
7. 7.
Chen, H.: Interesting series associated with central binomial coefficients, Catalan numbers and harmonic numbers. J. Integer Seq. 19(1), Article 16.1.5, 11 (2016)
8. 8.
Chu, W., Zheng, D.: Infinite series with harmonic numbers and central binomial coefficients. Int. J. Number Theory 5(3), 429–448 (2009)
9. 9.
Guillera, J.: More hypergeometric identities related to Ramanujan-type series. Ramanujan J. 32(1), 5–22 (2013)
10. 10.
Liu, H., Zhou, W., Ding, S.: Generalized harmonic number summation formulae via hypergeometric series and digamma functions. J. Differ. Equ. Appl. 23(7), 1204–1218 (2017)
11. 11.
Nicholson, M.: Quadratic transformations of hypergeometric function and series with harmonic numbers. ArXiv e-prints. arXiv:1801.02428 (2018)
12. 12.
Sofo, A.: General order Euler sums with multiple argument. J. Number Theory 189, 255–271 (2018)
13. 13.
Wan, J.G.F.: Random Walks, Elliptic Integrals and Related Constants. Ph.D. Thesis, University of Newcastle (2013)Google Scholar

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