Basic asymptotic estimates for powers of Wallis‘ ratios

Vito Lampret

doi:10.4067/S0719-06462021000300357

DOI:

https://doi.org/10.4067/S0719-06462021000300357

Abstract

For any \(a\in{\mathbb R}\), for every \(n\in{\mathbb N}\), and for \(n\)-th Wallis' ratio \(w_n:=\prod_{k=1}^n\frac{2k-1}{2k}\), the relative error \(r_{\,\!_0}(a,n):=\big(v_{\,\!_0}(a,n)-w_n^a\big)/w_n^a\) of the approximation \(w_n^a\approx v_{\,\!_0}(a,n):=(\pi n)^{-a/2} \) is estimated as \( \big|r_{\,\!_0}(a,n)\big| < \frac{1}{4n}\). The improvement \(w_n^a\approx v(a,n):=(\pi n)^{-a/2}\left(1-\frac{a}{8n}+\frac{a^2}{128n^2}\right)\) is also studied.

Keywords

approximation , asymptotic , estimate , inequality , power , Wallis‘ ratio

Vito Lampret University of Ljubljana, Ljubljana, 386 Slovenia, EU.

Pages: 357–368
Date Published: 2021-12-01
Vol. 23 No. 3 (2021)

H. Alzer, “Inequalities for the constants of Landau and Lebesgue”, J. Comput. Appl. Math., vol. 139, no. 2, pp. 215–230, 2002.

T. BuriÄ‡ and N. ElezoviÄ‡, “Bernoulli polynomials and asymptotic expansions of the quotient of gamma functions”, J. Comput. Appl. Math., vol. 235, no. 11, pp. 3315–3331, 2011.

T. BuriÄ‡ and N. ElezoviÄ‡, “New asymptotic expansions of the quotient of gamma functions”, Integral Transforms and Spec. Funct., vol. 23, no. 5, pp. 355–368, 2012.

C.-P. Chen and F. Qi, “The best bounds in Wallis‘ inequality”, Proc. Amer. Math. Soc., vol. 133, no. 2, pp. 397–401, 2005.

V. G. Cristea, “A direct approach for proving Wallis ratio estimates and an improvement of Zhang-Xu-Situ inequality”, Studia Univ. BabeÅŸ-Bolyai Math., vol. 60, no. 2, pp. 201–209, 2015.

S. Dumitrescu, “Estimates for the ratio of gamma functions using higher order roots”, Stud. Univ. BabeÅŸ-Bolyai Math., vol. 60, pp. 173–181, 2015.

N. ElezoviÄ‡, L. Lin and L. VukšiÄ‡, “Inequalities and asymptotic expansions of the Wallis sequence and the sum of the Wallis ratio”, J. Math. Inequal., vol. 7, no. 4, pp. 679–695, 2013.

N. ElezoviÄ‡, “Asymptotic expansions of gamma and related functions, binomial coefficient, inequalities and means”, J. Math. Inequal., vol. 9, no. 4, pp. 1001–1054, 2015.

T. Friedmann and C. R. Hagen, “Quantum mechanical derivation of the Wallis formula for Ï€”, J. Math. Phys., vol. 56, no. 11, 3 pages, 2015.

S. Guo, J.-G. Xu and F. Qi, “Some exact constants for the approximation of the quantity in the Wallis‘ formula”, J. Inequal. Appl., vol. 2013 , no. 67, 7 pages, 2013.

S. Guo, Q. Feng, Y.-Q. Bi and Q.-M. Luo; “A sharp two-sided inequality for bounding the Wallis ratio”, J. Inequal. Appl., vol. 2015, no. 43, 5 pages, 2015.

P. Haggstrom, Quantum mechanical derivation of the Wallis formula for Pi, https: //gotohaggstrom.com/QuantummechanicalderivationoftheWallisformulaforPi.pdf, 2020.

M. D. Hirschhorn, “Comments on the paper: “Wallis sequence estimated through the Euler- Maclaurin formula: even from the Wallis product Ï€ could be computed fairly accurately” by V. Lampret”, Austral. Math. Soc. Gaz., vol. 32, no. 3, pp. 194, 2005.

D. K. Kazarinoff, “On Wallis‘ formula”, Edinburgh Math. Notes, no. 40, pp. 19–21, 1956.

T. Koshy, Catalan numbers with applications, New York: Oxford University Press, 2009.

A. Laforgia and P. Natalini, “On the asymptotic expansion of a ratio of gamma functions”, J. Math. Anal. Appl., vol. 389, no. 4, pp. 833–837, 2012.

V. Lampret, “An asymptotic approximation of Wallis‘ sequence”, Cent. Eur. J. Math., vol. 10, no. 2, pp. 775–787, 2012.

V. Lampret, “Wallis‘ sequence estimated accurately using an alternating series”, J. Number Theory, vol. 172, pp. 256–269, 2017.

V. Lampret, “A simple asymptotic estimate of Wallis‘ ratio using Stirling‘s factorial formula”, Bull. Malays. Math. Sci. Soc., vol. 42, no. 6, pp. 3213–3221, 2019.

V. Lampret, “The perimeter of a flattened ellipse can be estimated accurately even from Maclaurin‘s series”, Cubo, vol. 21, no. 2, pp. 51–64, 2019.

L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics: Mechanics, 3ed, Oxford: Butterworth-Heinemann, 1986.

C. Mortici, “Sharp inequalities and complete monotonicity for the Wallis ratio”, Bull. Belg. Math. Math. Soc. Simon Stevin, vol. 17, no. 5, pp. 929–936, 2010.

C. Mortici, “A new method for establishing and proving new bounds for the Wallis ratio”, Math. Inequal. Appl., vol. 13, no. 4, pp. 803–815, 2010.

C. Mortici, “Refinements of Gurland‘s formula for pi”, Comput. Math. Appl., vol. 62, no. 6, pp. 2616–2620, 2011.

C. Mortici, “Sharp bounds of the Landau constants”, Math. Comp., vol. 80, no. 274, pp. 1011–1015, 2011.

C. Mortici, “Completely monotone functions and the Wallis ratio”, Appl. Math. Lett., vol. 25, no. 4, pp. 717–722, 2012.

C. Mortici and V. G. Cristea, “Estimates for Wallis‘ ratio and related functions”, Indian J. Pure Appl. Math., vol. 47, no. 3, pp. 437–447, 2016.

F. Qi and C. Mortici, “Some best approximation formulas and the inequalities for the Wallis ratio”, Appl. Math. Comput., vol. 253, pp. 363–368, 2015.

F. Qi, “An improper integral, the beta function, the Wallis ratio, and the Catalan numbers”, Probl. Anal. Issues Anal., vol. 7(25), no. 1, pp. 104–115, 2018.

D. V. SlaviÄ‡, “On inequalities for Î“(x + 1)/Î“(x + 1/2)”, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., no. 498–541, pp. 17–20, 1975.

J.-S. Sun and C.-M. Qu, “Alternative proof of the best bounds of Wallis‘ inequality”, Commun. Math. Anal., vol. 2, no. 1, pp. 23–27, 2007.

S. Wolfram, Mathematica, version 7.0, Wolfram Research, Inc., 1988–2009.

X.-M. Zhang, T. Q. Xu and L. B. Situ “Geometric convexity of a function involving gamma function and application to inequality theory”, JIPAM. J. Inequal. Pure Appl. Math., vol. 8, no. 1, art. 17, 9 pages, 2007.