Asymptotic approximation of central binomial coefficients with rigorous error bounds

@article{Brent2021AsymptoticAO,
  title={Asymptotic approximation of central binomial coefficients with rigorous error bounds},
  author={Richard P. Brent},
  journal={Open Journal of Mathematical Sciences},
  year={2021}
}
  • R. Brent
  • Published 17 August 2016
  • Mathematics, Computer Science
  • Open Journal of Mathematical Sciences
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. 
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ON THE ACCURACY OF ASYMPTOTIC APPROXIMATIONS TO THE LOG-GAMMA AND RIEMANN–SIEGEL THETA FUNCTIONS
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  • Mathematics
    Journal of the Australian Mathematical Society
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New lower bounds are proved on D(n) and R (n) in terms of d = n-h, where d is the order of a Hadamard matrix and h is maximal subject to $h \le n$.
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