Asymptotic approximation of central binomial coefficients with rigorous error bounds

Abstract

We show that a well-known asymptotic series for the logarithm of the central binomial coefficient is strictly enveloping in the sense of Pólya and Szegö, 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.

11 pages, 1 table; added more references in v3, minor corrections in v4

References in corpus (2)

• On asymptotic approximations to the log-Gamma and Riemann-Siegel theta functions
• Probabilistic lower bounds on maximal determinants of binary matrices

Cited by in corpus (1)

• On asymptotic approximations to the log-Gamma and Riemann-Siegel theta functions