Basic asymptotic estimates for powers of Wallis’ ratios

• Autores: Vito Lampret
• Localización: Cubo: A Mathematical Journal, ISSN 0716-7776, ISSN-e 0719-0646, Vol. 23, Nº. 3, 2021, págs. 357-368
• Idioma: inglés
• DOI: 10.4067/S0719-06462021000300357
• Enlaces
  • Texto completo
• Resumen
  • español

    RESUMEN Para cualquier a ∈ ℝ, para todo n ∈ ℕ, y para el n- ésimo cociente de Wallis de la aproximación se estima como También se estima la mejora .

  • English

    ABSTRACT For any a ∈ ℝ, for every n ∈ ℕ, and for n-th Wallis’ ratio of the approximation is estimated as The improvement is also studied.

• Referencias bibliográficas
  • Alzer, H.. (2002). Inequalities for the constants of Landau and Lebesgue. J. Comput. Appl. Math.. 139. 215
  • Burić, T.,Elezović, N.. (2011). Bernoulli polynomials and asymptotic expansions of the quotient of gamma functions. J. Comput. Appl. Math.....
  • Burić, T.,Elezović, N.. (2012). New asymptotic expansions of the quotient of gamma functions. Integral Transforms and Spec. Funct.. 23. 355
  • Chen, C.-P.,Qi, F.. (2005). The best bounds in Wallis’ inequality. Proc. Amer. Math. Soc.. 133. 397-401
  • Cristea, V. G.. (2015). A direct approach for proving Wallis ratio estimates and an improvement of Zhang-Xu-Situ inequality. Studia Univ....
  • Dumitrescu, S.. (2015). Estimates for the ratio of gamma functions using higher order roots. Stud. Univ. Babeş-Bolyai Math. 60. 173
  • Elezović, N.,Lin, L.,Vukšić, L.. (2013). Inequalities and asymptotic expansions of the Wallis sequence and the sum of the Wallis ratio. J....
  • Elezović, N.. (2015). Asymptotic expansions of gamma and related functions, binomial coefficient, inequalities and means. J. Math. Inequal.....
  • Friedmann, T.,Hagen, C. R.. (2015). Quantum mechanical derivation of the Wallis formula for π. J. Math. Phys.. 56. 3
  • Guo, S.,Xu, J.-G.,Qi, F.. (2013). Some exact constants for the approximation of the quantity in the Wallis’ formula. J. Inequal. Appl.. 2013....
  • Guo, S.,Feng, Q.,Bi, Y.-Q.,Luo, Q.-M.. (2015). A sharp two-sided inequality for bounding the Wallis ratio. J. Inequal. Appl.. 2015. 5
  • (2020). Quantum mechanical derivation of the Wallis formula for Pi.
  • Hirschhorn, M. D.. (2005). Comments on the paper: “Wallis sequence estimated through the Euler-Maclaurin formula: even from the Wallis product...
  • Kazarinoff, D. K.. (1956). On Wallis’ formula. Edinburgh Math. Notes. 19-21
  • Koshy, T.. (2009). Catalan numbers with applications. Oxford University Press. New York.
  • Laforgia, A.,Natalini, P.. (2012). On the asymptotic expansion of a ratio of gamma functions. J. Math. Anal. Appl.. 389. 833
  • Lampret, V.. (2012). An asymptotic approximation of Wallis’ sequence. Cent. Eur. J. Math.. 10. 775
  • Lampret, V.. (2017). Wallis’ sequence estimated accurately using an alternating series. J. Number Theory. 172. 256
  • Lampret, V.. (2019). A simple asymptotic estimate of Wallis’ ratio using Stirling’s factorial formula. Bull. Malays. Math. Sci. Soc.. 42....
  • Lampret, V.. (2019). The perimeter of a flattened ellipse can be estimated accurately even from Maclaurin’s series. Cubo. 21. 51-64
  • Landau, L. D.,Lifshitz, E. M.. (1986). Course of Theoretical Physics: Mechanics. 3. Butterworth-Heinemann. Oxford.
  • Mortici, C.. (2010). Sharp inequalities and complete monotonicity for the Wallis ratio. Bull. Belg. Math. Math. Soc. Simon Stevin. 17. 929
  • Mortici, C.. (2010). A new method for establishing and proving new bounds for the Wallis ratio. Math. Inequal. Appl.. 13. 803
  • Mortici, C.. (2011). Refinements of Gurland’s formula for pi. Comput. Math. Appl.. 62. 2616
  • Mortici, C.. (2011). Sharp bounds of the Landau constants. Math. Comp.. 80. 1011
  • Mortici, C.. (2012). Completely monotone functions and the Wallis ratio. Appl. Math. Lett.. 25. 717
  • Mortici, C.,Cristea, V. G.. (2016). Estimates for Wallis’ ratio and related functions. Indian J. Pure Appl. Math.. 47. 437
  • Qi, F.,Mortici, C.. (2015). Some best approximation formulas and the inequalities for the Wallis ratio. Appl. Math. Comput.. 253. 363
  • Qi, F.. (2018). An improper integral, the beta function, the Wallis ratio, and the Catalan numbers. Probl. Anal. Issues Anal.. 7. 104
  • Slavić, D. V.. (1975). On inequalities for Γ(x + 1)/Γ(x + 1/2). Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz.,. 17-20
  • Sun, J.-S.,Qu, C.-M.. (2007). Alternative proof of the best bounds of Wallis’ inequality. Commun. Math. Anal.. 2. 23
  • Wolfram, S.. (2009). Mathematica, version 7.0. Wolfram Research, Inc.
  • Zhang, X.-M.,Xu, T. Q.,Situ, L. B.. (2007). Geometric convexity of a function involving gamma function and application to inequality theory....