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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.32 no.2 Antofagasta May 2013

http://dx.doi.org/10.4067/S0716-09172013000200006 

Proyecciones Journal of Mathematics Vol. 32, No 2, pp. 173-181, June 2013. Universidad Católica del Norte Antofagasta - Chile

An approximation formula for n!

Necdet Batir

Nev§eh1r University, Nev§eh1r, Turkey


ABSTRACT

We prove the following very accurate approximation formula for the factorial function:

This gives better results than the following approximation formula

which is established by the author [5] and C. Mortici [16] independently, and gives similar results with

which is established by C. Mortici in his very new paper [8].

Subjclass [2000] Primary : 33B15; Secondary: 26D07, 11Y60.

Keywords : Gamma function, Stirling formula, Euler-Mascheroni constant, harmonic numbers, inequalities, digamma function.


REFERENCES

[I] H. Alzer, On some inequalities for the gamma and psi function, Math.Comput., 66, 217, pp. 373-389, (1997).

[2] H. Alzer, On Ramanujan's double inequality for the gamma function,Bull. London Math. Soc. 5, 35, pp. 601-607, (2003).         [ Links ]

[3] N. Batir, Inequalities for the gamma function, Arch. Math(Basel), 91,pp. 554-563, (2008).         [ Links ]

[4] N. Batir, Sharp inequalities for factorial n, Proyeccioes, 1, 27, pp. 97102, (2008).

[5] N. Batir, Very accurate approximations for the factorial function, J.Math. Inequal., 3, 4, pp. 335-344, (2010).         [ Links ]

[6] N. Batir, Improving Stirling formula, Math. Commun., 16, No. 1, pp.105-114, (2011).         [ Links ]

[7] L. Bauer, Remark on Stirling's formulas and on approximations for the double factorial, Mathematical Intelligences, 28, 2, pp. 10-21, (2006).         [ Links ]

[8] E. A. Karatsuba, On the asymptotic representation of the Euler gamma function by Ramanujan, J. Comput. Appl. Math., 135, 2001, 225-240.         [ Links ]

[9] C. Mortici, Ramanujan's estimate for the gamma function via mono-tonicity arguments, Ramanujan J, DOI.10.1007/s11139-010-9265-y,(2011).         [ Links ]

[10] C. Mortici, A new Stirling series as continued fraction, Numerical Algorithms, 56, no. 1, pp. 17-26, (2011).         [ Links ]

[II] C. Mortici, Accurate estimates of the gamma function involving the psi function, Numer. Funct. Anal. Optim., 32, no. 4, pp. 469-476, (2011).

[12] C. Mortici, New sharp inequalities for approximating the factorial function and the digamma function, Miskolc Math., 11, no. 1, pp. 79-86, (2010).         [ Links ]

[13] C. Mortici, Asymptotic expansions of the generalized Stirling approximation, Mathe. Comput. Model., 52, no. 9-10, pp. 1867-1868, (2010).         [ Links ]

[14] C. Mortici, Estimating gamma function in terms of digamma function,Math. Comput. Model., 52, no. 5-6, pp. 942-946, (2010).         [ Links ]

[15] C. Mortici, On the Stirling expansion into negative powers of a triangular numbers, Math. Commun., 15, no. 2, pp. 359-364, (2010).         [ Links ]

[16] C. Mortici, Sharp inequalities related to Gosper's Formula, Comptes Rendus Mathematique, 348, no. 3-4, pp. 137-140, (2010).         [ Links ]

[17] C. Mortici, A class of integral approximations for the factorial function, Computers and Mathematics with Applications, 59, no. 6, pp. 20532058, (2010).         [ Links ]

[18] C. Mortici, New improvements of the Stirling formula , Appl. Math.Comput., 217, no. 2, pp. 699-704, (2010).         [ Links ]

[19] C. Mortici, New approximations of the gamma function in terms of the digamma function, Applied Mathematics Letters, 23, no. 1, pp. 97-100,(2010).         [ Links ]

[20] C. Mortici, An ultimate extremely accurate formula for approximation of the factorial function, Arch. Math. (Basel), 93, no. 1, pp. 37-45,(2009).         [ Links ]

 

Necdet Batir
Department of Mathematics, Faculty of Science and Arts, Nevsehir University,
Nevsehir, Turkey

e-mail : nbatir@hotmail.com

Received : July 2012. Accepted : April 2013

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