Abstract
In this paper, we present a new formula for approximating the Wallis ratio. This new formula is very fast in comparison with other classical or recently discovered asymptotic series. Next, inequalities related to this new formula and asymptotic series are established. Finally, for demonstrating the superiority of our new series, some numerical computations are provided.
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References
Alzer H.: On some inequalities for the gamma and psi functions. Math. Comput. 66(217), 373–389 (1997)
Chen C.-P., Qi F.: The best bounds in Wallis’ inequality. Proc. Am. Math. Soc. 133(2), 397–401 (2005)
Guo, S., Xu, J.-G., Qi, F.: Some exact constants for the approximation of the quantity in the Wallis’ formula. J. Inequal. Appl. 2013, 67 (2013)
Mortici C.: Product approximations via asymptotic integration. Am. Math. Mon. 117, 434–441 (2010)
Mortici C.: A new Stirling series as continued fraction. Numer. Algorithms 56(1), 17–26 (2011)
Mortici C.: A continued fraction approximation of the gamma function. J. Math. Anal. Appl. 402(2), 405–410 (2013)
Mortici, C.: A new fast asymptotic series for the gamma function. J. Ramanujan. doi:10.1007/s11139-014-9589-0 (2015, online)
Qi F., Mortici C.: Some best approximation formulas and inequalities for the Wallis ratio. App. Math. Comput. 253, 363–368 (2015)
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Liu, X., Lu, D. & Song, L. Some Quicker Approximations and Inequalities of the Wallis Ratio by Continued Fraction. Results. Math. 70, 325–335 (2016). https://doi.org/10.1007/s00025-015-0509-4
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DOI: https://doi.org/10.1007/s00025-015-0509-4