×

On Stirling’s formula remainder. (English) Zbl 1338.33004

Summary: Let the sequence \(r_n\) be defined by \[ n! = \sqrt{2 \pi n} (n / e)^n e^{r_n}. \] We establish new estimates for Stirling’s formula remainder \(r_n\). This improves some known results. Let \(\theta(x)\) be defined by the relation: \[ \Gamma(x + 1) = \sqrt{2 \pi}(x / e)^x e^{\theta(x) /(12 x)}. \] We prove that \(\theta''(x)\) is completely monotonic on \((0, \infty)\). This implies the result given by Mortici, who proved that \(- x^{- 1} \theta'''(x)\) is completely monotonic on \((0, \infty)\).

MSC:

33B15 Gamma, beta and polygamma functions
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abramowitz, M.; Stegun, I. A., Handbook of mathematical functions with formulas, graphs, and mathematical tables, (Applied Mathematics Series, vol. 55 (1972), National Bureau of Standards: National Bureau of Standards Washington, DC), ninth printing · Zbl 0543.33001
[2] Alzer, H., On some inequalities for the gamma and psi functions, Math. Comp., 66, 373-389 (1997) · Zbl 0854.33001
[3] Batir, N., Very accurate approximations for the factorial function, J. Math. Inequal., 4, 335-344 (2010) · Zbl 1196.33002
[4] Beesack, P. R., Improvement of Stirling’s formula by elementary methods, Univ. Beograd. Publ. Elektrotehn. Fak. Set. Mat. Fiz., 274-301, 17-21 (1969) · Zbl 0203.36301
[5] Chen, C.-P.; Batir, N., Some inequalities and monotonicity properties associated with the gamma and psi functions, Appl. Math. Comput., 218, 8217-8225 (2012) · Zbl 1250.33003
[6] Chen, C.-P.; Mortici, C., New sequence converging towards the Euler-Mascheroni constant, Comput. Math. Appl., 64, 391-398 (2012) · Zbl 1252.33002
[7] Dubourdieu, J., Sur un théorème de M.S. Bernstein relatif á la transformation de Laplace-Stieltjes, Compos. Math., 7, 96-111 (1939), (in French) · JFM 65.0473.02
[8] van Haeringen, H., Completely monotonic and related functions, J. Math. Anal. Appl., 204, 389-408 (1996) · Zbl 0889.26008
[9] Hummel, P. M., A note on Stirling’s formula, Am. Math. Mon., 47, 97-99 (1940)
[10] Mortici, C., An ultimate extremely accurate formula for approximation of the factorial function, Arch. Math. (Basel), 93, 37-45 (2009) · Zbl 1186.40004
[11] Mortici, C., Best estimates of the generalized Stirling formula, Appl. Math. Comput., 215, 4044-4048 (2010) · Zbl 1186.33003
[12] Mortici, C., Sharp inequalities related to Gosper’s formula, C.R. Math. Acad. Sci. Paris, 348, 137-140 (2010) · Zbl 1186.33004
[13] Mortici, C., Product approximations via asymptotic integration, Am. Math. Mon., 117, 434-441 (2010) · Zbl 1214.40002
[14] Mortici, C., New improvements of the Stirling formula, Appl. Math. Comput., 217, 699-704 (2010) · Zbl 1202.33004
[15] Mortici, C., On the monotonicity and convexity of the remainder of the Stirling formula, Appl. Math. Lett., 24, 869-871 (2011) · Zbl 1216.33008
[16] Mortici, C., A continued fraction approximation of the gamma function, J. Math. Anal. Appl., 402, 405-410 (2013) · Zbl 1333.40001
[17] Nanjundiah, T. S., Note on Stirling’s formula, Am. Math. Mon., 66, 701-703 (1959) · Zbl 0091.06401
[18] Robbins, H., A remark on Stirling’s formula, Am. Math. Mon., 62, 26-29 (1955) · Zbl 0068.05404
[19] Shi, X.; Liu, F.; Hu, M., A new asymptotic series for the gamma function, J. Comput. Appl. Math., 195, 134-154 (2006) · Zbl 1098.33002
[20] Whittaker, E. T.; Watson, G. N., A Course in Modern Analysis (1990), Cambridge University Press: Cambridge University Press Cambridge, England
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.