Numerical Algorithms

, Volume 56, Issue 1, pp 17–26 | Cite as

A new Stirling series as continued fraction

Original Paper

Abstract

We introduce the following new Stirling series
$$ n!\sim \sqrt{2\pi n}\left( \frac{n}{e}\right) ^{n}\exp \frac{1}{12n+\frac{ \frac{2}{5}}{n+\frac{\frac{53}{210}}{n+\frac{\frac{195}{371}}{n+\frac{\frac{ 22,\!999}{22,\!737}}{n+\ddots}}}}}, $$
as a continued fraction, which is faster than the classical Stirling series.

Keywords

Stirling formula Rate of convergence Approximations Asymptotic expansions 

Mathematics Subject Classifications (2000)

33B15 41A10 42A16 

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References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Applied Mathematical Series, 55, 9th Printing. Dover, New York (1972)Google Scholar
  2. 2.
    Alzer, H.: On some inequalities for the gamma and psi functions. Math. Comput. 66(217), 373–389 (1997)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Batir, N.: Sharp inequalities for factorial n. Proyecciones 27(1), 97–102 (2008)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Beesack, P.R.: Improvements of Stirling’s formula by elementary methods. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 274–301, 17–21 (1969)MathSciNetGoogle Scholar
  5. 5.
    Bracken, P.: Properties of certain sequences related to Stirling’s approximation for the gamma function. Expo. Math. 21, 171–178 (2003)MATHMathSciNetGoogle Scholar
  6. 6.
    Burnside, W.: A rapidly convergent series for logN!. Messenger Math. 46, 157–159 (1917)Google Scholar
  7. 7.
    Cesaró, E.: Elementares Lehrbuch der Algebraischen Analysis und der Infinitesimalrechnung, p. 154. Springer, Leipzig (1922)Google Scholar
  8. 8.
    Dominici, D.: Variations on a Theme by James Stirling. http://arxiv.org/abs/math/0603007
  9. 9.
    Feller, W.: Stirling’s Formula. §2.9 in An Introduction to Probability Theory and Its Applications, 3rd edn., vol. 1, pp. 50–53. Wiley, New York (1968)Google Scholar
  10. 10.
    Fowler, D.: The factorial function: Stirling’s formula. Math. Gaz. 84, 42–50 (2000)CrossRefGoogle Scholar
  11. 11.
    Gosper, R.W.: Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. U. S. A. 75, 40–42 (1978)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Hummel, P.M.: A note on Stirling’s formula. Am. Math. Mon. 47(2), 1940 (1940)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Maria, A.J.: A remark on Stirling’s formula. Am. Math. Mon. 72, 1096–1098 (1965)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Mortici, C.: Product approximations via asymptotic integration. Am. Math. Mon. 117(5), 434–441 (2010)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Mortici, C.: An ultimate extremely accurate formula for approximation of the factorial function. Arch. Math. (Basel) 93(1), 37–45 (2009)MATHMathSciNetGoogle Scholar
  16. 16.
    Mortici, C.: New approximations of the gamma function in terms of the digamma function. Appl. Math. Lett. 23(1), 97–100 (2010)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Mortici, C.: New sharp bounds for gamma and digamma functions. An. Ştiinţ. Univ. A. I. Cuza Iaşi Ser. N. Matem. vol. 56, no. 2 (2010a, in press)Google Scholar
  18. 18.
    Mortici, C.: Completely monotonic functions associated with gamma function and applications. Carpath. J. Math. 25(2), 186–191 (2009)Google Scholar
  19. 19.
    Mortici, C.: The proof of Muqattash-Yahdi conjecture. Math. Comput. Model. (2010b, in press). doi:10.1016/j.mcm.2009.12.030 Google Scholar
  20. 20.
    Mortici, C.: Monotonicity properties of the volume of the unit ball in ℝ n. Optimization Lett. (2010). doi:10.1007/s11590-009-0173-2 MathSciNetGoogle Scholar
  21. 21.
    Mortici, C.: Sharp inequalities related to Gosper’s formula. C. R. Math. Acad. Sci. Paris (2010c, in press). doi:10.1016/j.crma.2009.12.016 MathSciNetGoogle Scholar
  22. 22.
    Mortici, C.: A class of integral approximations for the factorial function. Comput. Math. Appl. (2010d, in press). doi:10.1016/j.camwa.2009.12.010 Google Scholar
  23. 23.
    Mortici, C.: Best estimates of the generalized Stirling formula. Appl. Math. Comput. 215(11), 4044–4048 (2010)MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Mortici, C.: Very accurate estimates of the polygamma functions. Asymptot. Anal. (2010). doi:10.3233/ASY2010-0983 MathSciNetGoogle Scholar
  25. 25.
    Mortici, C.: Improved convergence towards generalized Euler-Mascheroni constant. Appl. Math. Comput. 215(9), 3443–3448 (2010)MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Mortici, C.: A coincidence degree for bifurcation problems. Nonlinear Anal. 53(5), 715–721 (2003)MATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Mortici, C.: A quicker convergence toward the γ constant with the logarithm term involving the constant e. Carpath. J. Math. 26(1) (2010e, in press)Google Scholar
  28. 28.
    Mortici, C.: Optimizing the rate of convergence in some new classes of sequences convergent to Euler’s constant. Anal. Appl. (Singap.) 8(1), 99–107 (2010)MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    O’Connor, J., Robertson, E.F.: James Stirling, MacTutor History of Mathematics Archive (2010)Google Scholar
  30. 30.
    Qi, F.: Three classes of logarithmically completely monotonic functions involving gamma and psi functions. Integral Transforms Spec. Funct. 18(7), 503–509 (2007)MATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. Intr. by G. E. Andrews. Narosa Publishing House, New Delhi (1988)Google Scholar
  32. 32.
    Robbins, H.: A Remark of Stirling’s Formula. Am. Math. Mon. 62, 26–29 (1955)MATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Stirling, J.: Methodus differentialis, sive tractatus de summation et interpolation serierum infinitarium. London (1730). English translation by J. Holliday, The Differential Method: A Treatise of the Summation and Interpolation of Infinite SeriesGoogle Scholar
  34. 34.
    Temme, N.M.: Uniform asymptotic expansions of integrals: a selection of problems.break J. Comput. Appl. Math. 65, 395–417 (1995)MATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Temme, N.M.: Special Functions: An Introduction to the Classical Functions of Mathematical Physics. John Wiley and Sons, New York (1996)MATHGoogle Scholar
  36. 36.
    Tweddle, I.: Approximating n!. Historical origins and error analysis. Am. J. Phys. 52, 487 (1984)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  1. 1.Department of MathematicsValahia University of TârgovişteTârgovişteRomania

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