Asymptotic approximation of central binomial coefficients with rigorous error bounds

TL;DR

This paper proves that a classical asymptotic series for the central binomial coefficient provides strict bounds with errors of predictable sign and magnitude, and explores related series for special functions.

Contribution

It establishes rigorous error bounds for the asymptotic series of the central binomial coefficient and related functions, enhancing understanding of their accuracy.

Findings

Asymptotic series is strictly enveloping with predictable error signs.

Error bounds are explicitly tied to the next term in the series.

Connections to Binet's function and the Riemann-Siegel theta function are explored.

Abstract

We show that a well-known asymptotic series for the logarithm of the central binomial coefficient is strictly enveloping in the sense of P\'olya and Szeg\"o, so the error incurred in truncating the series is of the same sign as the next term, and is bounded in magnitude by that term. We consider closely related asymptotic series for Binet's function, for $\ln\Gamma(z+1/2)$, and for the Riemann-Siegel theta function, and make some historical remarks.