Asymptotic approximation of central binomial coefficients with rigorous error bounds
Abstract
We show that a wellknown 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 RiemannSiegel theta function, and make some historical remarks.
 Publication:

arXiv eprints
 Pub Date:
 August 2016
 arXiv:
 arXiv:1608.04834
 Bibcode:
 2016arXiv160804834B
 Keywords:

 Mathematics  Numerical Analysis;
 Mathematics  Classical Analysis and ODEs;
 Mathematics  Combinatorics;
 05A10;
 11B65 (Primary);
 33B15;
 41A60 (Secondary)
 EPrint:
 11 pages, 1 table