On the Limiting Structure of Some Central Binomial Evaluations

Abstract

We examine series of the form $$\sum_{n=0}^\infty {2n \choose n}^{-1} \frac{(4x^2)^n}{2n+2m+1} \textrm{and} \sum_{n=0}^\infty {2n \choose n}^{-1} \frac{(-4x^2)^n}{2n+2m+1} .$$ In each case, there is an evaluation of the form $(p_m (x) f(x) - q_m (x))/x^{2m}$ where $f(x)$ is a transcendental function and $p_m (x)$ and $q_m (x)$ are polynomials with rational coefficients. We prove that for $\vert x \vert \lt 1$, $$\lim_{m \to \infty} \frac{q_m (x)}{p_m (x)} = f(x) .$$ From this result, we derive recurrences for $\pi$ and for various logarithms.