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On Burnside type approximation for the gamma function

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Abstract

In the paper, by means of the monotonicity rule for the ratio of two Laplace transform we determine the best \(a,b>0\) such that the double inequality

$$\begin{aligned} \exp \left( -\frac{x}{24x^{2}+b}\right)<\frac{\Gamma \left( x+1/2\right) }{ \sqrt{2\pi }\left( x/e\right) ^{x}}<\exp \left( -\frac{x}{24x^{2}+a}\right) \end{aligned}$$

holds for \(x>x_{0}\ge 0\). Furthermore, we give some completely monotonic functions involving gamma function, and establish an asymptotic expansion of \(\Gamma \left( x+1/2\right) \). Finally, an open problem is proposed.

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References

  1. Burnside, W.: A rapidly convergent series for \(\log N!\). Messenger Math. 46, 157–159 (1917)

    MATH  Google Scholar 

  2. Luke, Y.L.: The Special Functions and Their Approximations, vol. I. Academic Press, New York (1969)

    MATH  Google Scholar 

  3. Yang, Z.: Approximations for certain hyperbolic functions by partial sums of their Taylor series and completely monotonic functions related to gamma function. J. Math. Anal. Appl. 441, 549–564 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Smith, W.D.: The gamma function revisited. http://schule.bayernport.com/gamma/gamma05.pdf (2006)

  5. Mortici, C.: On the gamma function approximation by burnside. Appl. Math. E-Notes 11, 274–277 (2011)

    MathSciNet  MATH  Google Scholar 

  6. Lu, D., Wang, X.: A new asymptotic expansion and some inequalities for the gamma function. J. Number Theory 140, 314–323 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Yang, Z., Chu, Y.: Asymptotic formulas for gamma function with applications. Appl. Math. Comput. 270, 665–680 (2015)

    MathSciNet  MATH  Google Scholar 

  8. Yang, Z., Tian, J.: Monotonicity and inequalities for the gamma function. J. Inequal. Appl. 2017, 317 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Yang, Z., Tian, J.: An accurate approximation formula for gamma function. J. Inequal. Appl. 2018, 56 (2018)

    Article  MathSciNet  Google Scholar 

  10. Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. Springer, Berlin (1988)

    MATH  Google Scholar 

  11. Shi, X., Liu, F., Hu, M.: A new asymptotic series for the Gamma function. J. Comput. Appl. Math. 195, 134–154 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Yang, Z., Tian, J.: A comparison theorem for two divided differences and applications to special functions. J. Math. Anal. Appl. 464, 580–595 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Yang, Z., Tian, J.: A class of completely mixed monotonic functions involving gamma function with applications. Proc. Am. Math. Soc. 146, 4707–4721 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Yang, Z., Tian, J.: Two asymptotic expansions for gamma function developed by Windschitl’s formula. Open Math. 16, 1048–1060 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Batir, N.: Inequalities for the gamma function. Arch. Math. 91, 554–563 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mortici, C.: An ultimate extremely accurate formula for approximation of the factorial function. Arch. Math. 93, 37–45 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nemes, G.: New asymptotic expansion for the Gamma function. Arch. Math. (Basel) 95, 161–169 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mortici, C.: New improvements of the Stirling formula. Appl. Math. Comput. 217, 699–704 (2010)

    MathSciNet  MATH  Google Scholar 

  19. Mortici, C.: Ramanujan formula for the generalized Stirling approximation. Appl. Math. Comput. 217, 2579–2585 (2010)

    MathSciNet  MATH  Google Scholar 

  20. Mortici, C.: Improved asymptotic formulas for the gamma function. Comput. Math. Appl. 61, 3364–3369 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mortici, C.: Further improvements of some double inequalities for bounding the gamma function. Math. Comput. Model. 57, 1360–1363 (2013)

    Article  MathSciNet  Google Scholar 

  22. Feng, L., Wang, W.: Two families of approximations for the gamma function. Numer. Algorithms 64, 403–416 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lu, D.: A new sharp approximation for the Gamma function related to Burnside’s formula. Ramanujan J. 35, 121–129 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lu, D., Song, L., Ma, C.: Some new asymptotic approximations of the gamma function based on Nemes’ formula, Ramanujan’s formula and Burnside’s formula. Appl. Math. Comput. 253, 1–7 (2015)

    MathSciNet  MATH  Google Scholar 

  25. Chen, C.: A more accurate approximation for the gamma function. J. Number Theory 164, 417–428 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yang, Z., Tian, J.: Monotonicity and sharp inequalities related to gamma function. J. Math. Inequal. 12(1), 1–22 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  27. Alzer, H.: On some inequalities for the gamma and psi functions. Math. Comput. 66, 373–389 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhao, J.-L., Guo, B.-N., Qi, F.: A refinement of a double inequality for the gamma function. Publ. Math. Debrecen 80, 333–342 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Guo, B.-N., Zhang, Y.-J., Qi, F.: Refinements and sharpenings of some double inequalities for bounding the gamma function. J. Inequal. Pure Appl. Math. 9(1), Article 17 (2008)

    MathSciNet  MATH  Google Scholar 

  30. Qi, F., Yang, Q., Li, W.: Two logarithmically completely monotonic functions connected with gamma function. Integral Transforms Spec. Funct. 17(7), 539–542 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Qi, F.: Bounds for the ratio of two gamma functions. J. Inequal. Appl. 2010, Art. ID 493058 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Qi, F., Chen, Ch-P: A complete monotonicity property of the gamma function. J. Math. Anal. Appl. 296(2), 603–607 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  33. Weisstein, E.W.: Laplace transform, from MathWorld—a Wolfram web resource. http://mathworld.wolfram.com/LaplaceTransform.html

  34. Yang, Z., Chu, Y., Wang, M.: Monotonicity criterion for the quotient of power series with applications. J. Math. Anal. Appl. 428, 587–604 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Belzunce, F., Ortega, E., Ruiz, J.M.: On non-monotonic ageing properties from the Laplace transform, with actuarial applications. Insur. Math. Econ. 40, 1–14 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Xia, F., Yang, Z., Chu, Y.: A new proof for the monotonicity criterion of the quotient of two power series on the infinite interval. Pac. J. Appl. Math. 7, 97–101 (2015)

    MathSciNet  MATH  Google Scholar 

  37. Yang, Z., Tian, J.: Sharp inequalities for the generalized elliptic integrals of the first kind. Ramanujan J. 48(1), 91–116 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  38. Yang, Z.: A new way to prove L’Hospital monotone rules with applications. arXiv:1409.6408 [math.CA]

  39. Yang, Z., Tian, J.: Monotonicity rules for the ratio of two Laplace transforms with applications. J. Math. Anal. Appl. 470(2), 821–845 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  40. Bernstein, S.: Sur les fonctions absolument monotones. Acta Math. 52, 1–66 (1929)

    Article  MathSciNet  MATH  Google Scholar 

  41. Widder, D.V.: Necessary and sufficient conditions for the representation of a function as a Laplace integral. Trans. Am. Math. Soc. 33, 851–892 (1931)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by the Fundamental Research Funds for the Central Universities (No. 2015ZD29) and the Higher School Science Research Funds of Hebei Province of China (No. Z2015137).

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Authors and Affiliations

  1. College of Science and Technology, North China Electric Power University, Ruixiang Street 282, Baoding, 071051, People’s Republic of China

    Zhen-Hang Yang & Jing-Feng Tian

  2. Department of Science and Technology, State Grid Zhejiang Electric Power Company Research Institute, Hangzhou, 310014, People’s Republic of China

    Zhen-Hang Yang

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  1. Zhen-Hang Yang
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  2. Jing-Feng Tian
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Yang, ZH., Tian, JF. On Burnside type approximation for the gamma function. RACSAM 113, 2665–2677 (2019). https://doi.org/10.1007/s13398-019-00651-2

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