Comptes Rendus
Mathematical Analysis/Numerical Analysis
Sharp inequalities related to Gosper's formula
[Inégalités précises liées à la formule de Gosper]
Comptes Rendus. Mathématique, Volume 348 (2010) no. 3-4, pp. 137-140.

Le but de cette Note est de construire un nouveau type de série de Stirling, étendant la formule de Gosper pour les grandes factorielles. Nous établissons de nouvelles inégalités précises pour les fonctions gamma et digamma. Enfin, nous indiquons des calculs numériques qui démontre la supériorité de notre nouvelle série sur la série classique de Stirling.

The purpose of this Note is to construct a new type of Stirling series, which extends the Gosper's formula for big factorials. New sharp inequalities for the gamma and digamma functions are established. Finally, numerical computations which demonstrate the superiority of our new series over the classical Stirling's series are given.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.12.016

Cristinel Mortici 1

1 Valahia University of Târgovişte, Department of Mathematics, Bd. Unirii 18, 130082 Târgovişte, Romania
@article{CRMATH_2010__348_3-4_137_0,
     author = {Cristinel Mortici},
     title = {Sharp inequalities related to {Gosper's} formula},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {137--140},
     publisher = {Elsevier},
     volume = {348},
     number = {3-4},
     year = {2010},
     doi = {10.1016/j.crma.2009.12.016},
     language = {en},
}
TY  - JOUR
AU  - Cristinel Mortici
TI  - Sharp inequalities related to Gosper's formula
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 137
EP  - 140
VL  - 348
IS  - 3-4
PB  - Elsevier
DO  - 10.1016/j.crma.2009.12.016
LA  - en
ID  - CRMATH_2010__348_3-4_137_0
ER  - 
%0 Journal Article
%A Cristinel Mortici
%T Sharp inequalities related to Gosper's formula
%J Comptes Rendus. Mathématique
%D 2010
%P 137-140
%V 348
%N 3-4
%I Elsevier
%R 10.1016/j.crma.2009.12.016
%G en
%F CRMATH_2010__348_3-4_137_0
Cristinel Mortici. Sharp inequalities related to Gosper's formula. Comptes Rendus. Mathématique, Volume 348 (2010) no. 3-4, pp. 137-140. doi : 10.1016/j.crma.2009.12.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.12.016/

[1] Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (M. Abramowitz; I.A. Stegun, eds.), Dover, New York, 1972

[2] G. Anderson; S. Qiu A monotoneity property of the gamma function, Proc. Amer. Math. Soc., Volume 125 (1997) no. 11, pp. 3355-3362

[3] L. Gordon A stochastic approach to the gamma function, Amer. Math. Monthly, Volume 101 (1994) no. 9, pp. 858-864

[4] R.W. Gosper Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. USA, Volume 75 (1978), pp. 40-42

[5] B.-N. Guo; F. Qi An algebraic inequality II, RGMIA Res. Rep. Coll. 4, Volume 1 (2001) no. 8, pp. 55-61 http://rgmia.vu.edu.au/v4n1.html (available online at)

[6] J.S. Martins Arithmetic and geometric means, an applications to Lorentz sequence spaces, Math. Nachr., Volume 139 (1988), pp. 281-288

[7] H. Minc; L. Sathre Some inequalities involving (n!)1/r, Proc. Edinburgh Math. Soc., Volume 14 (1964/65), pp. 41-46

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

[9] C. Mortici Complete monotonic functions associated with gamma function and applications, Carpathian J. Math., Volume 25 (2009) no. 2, pp. 186-191

[10] C. Mortici Product approximations via asymptotic integration, Amer. Math. Monthly, Volume 117 (2010) no. 5, pp. 434-441

[11] C. Mortici New approximations of the gamma function in terms of the digamma function, Appl. Math. Lett., Volume 23 (2010) no. 1, pp. 97-100

[12] C. Mortici, New sharp bounds for gamma and digamma functions, An. Ştiinţ. Univ. A. I. Cuza Iaşi Ser. N. Matem. 56 (2) (2010), in press

[13] C. Mortici Optimizing the rate of convergence in some new classes of sequences convergent to Euler's constant, Anal. Appl. (Singap.), Volume 8 (2010) no. 1, pp. 99-107

[14] F. Qi Three classes of logarithmically completely monotonic functions involving gamma and psi functions, Integral Transforms Spec. Funct., Volume 18 (2007) no. 7, pp. 503-509

Cité par Sources :

Commentaires - Politique