A Generalized Apery Series

TL;DR

This paper evaluates the inverse central binomial series for positive integers and explores its asymptotic behavior, revealing a special relation to pi at z=2, extending classical results by Apéry and Lehmer.

Contribution

It provides a generalized evaluation of the inverse central binomial series for any positive integer k and analyzes its asymptotics, highlighting the unique connection to pi at z=2.

Findings

Series evaluated for positive integers k

Asymptotic behavior characterized for large k

Special relation to pi at z=2

Abstract

The inverse central binomial series {equation}S_k(z)=\sum_{n=1}^{\infty}\frac{n^k z^n}{\binom{2n}{n}}{equation} popularized by Ap\'ery 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 $\pi$.