Abstract
In this note, using the well-known series representation for the Clausen function, we also provide some new representations of Apery’s constant \(\zeta (3)\). In addition, by an idea from De Amo et al. (Proc Am Math Soc 139:1441–1444, 2011) we derive some new rational series representations involving even zeta values and central binomial coefficients. These formulas are expressed in terms of odd and even values of the Riemann zeta function and odd values of the Dirichlet beta function. In particular cases, we recover some well-known series representations of \(\pi \).
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Acknowledgements
We would like to thank Piotr Hajlasz, Bogdan Ion, George Sparling and William C. Troy for some fruitful conversations which led to an improvement of the present paper.
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Lupu, C., Orr, D. Series representations for the Apery constant \(\zeta (3)\) involving the values \(\zeta (2n)\). Ramanujan J 48, 477–494 (2019). https://doi.org/10.1007/s11139-018-0081-0
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DOI: https://doi.org/10.1007/s11139-018-0081-0