Asymptotic approximation of central binomial coefficients with rigorous error bounds
Copyright policy )Asymptotic approximation of central binomial coefficients with rigorous error bounds
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 \(\ln\Gamma(z+\frac12)\), and for the Riemann-Siegel theta function, and make some historical remarks.
- Australian National University Australia
05A10, 11B65 (Primary), 33B15, 41A60 (Secondary), Mathematics - Classical Analysis and ODEs, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Mathematics - Combinatorics, Mathematics - Numerical Analysis, Numerical Analysis (math.NA), Combinatorics (math.CO)
05A10, 11B65 (Primary), 33B15, 41A60 (Secondary), Mathematics - Classical Analysis and ODEs, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Mathematics - Combinatorics, Mathematics - Numerical Analysis, Numerical Analysis (math.NA), Combinatorics (math.CO)
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