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A Generalized Apéry Series M. L. Glasser
Department of Physics
Clarkson University
Postdam, NY 13699-5820
USA
mailto:laryg@clarkson.edularyg@clarkson.edu

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Abstract:

The inverse central binomial series

$$S_k(z)=\sum_{n=1}^{\infty}\frac{n^k z^n}{\binom{2n}{n}},$$

popularized by Apéry and Lehmer is evaluated for positive integers $k$ along with the asymptotic behavior for large $k$. It is found that value $z=2$, as commented on by D. H. Lehmer, provides a unique relation to $\pi$.