Abstract
In the paper, the authors find integral representations, complete monotonicity, limits, and other properties of remainders of the Binet and Stirling formulas for the gamma function and their derivatives. These properties strengthen almost all results in three papers published in the Journal of Computational and Applied Mathematics, Applied Mathematics Letters, and Applied Mathematics and Computation in the years 2006, 2011, and 2014 by seven mathematicians. The proofs in the paper unify and are simpler than those in the three papers.
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Berg, C.: Integral representation of some functions related to the gamma function. Mediterr. J. Math. 1(4), 433–439 (2004). doi:10.1007/s00009-004-0022-6
Guo, S.-B., Ma, W.-B., Pradeep, B.G.S.A.: Complete characterizations of the gamma function. Appl. Math. Comput. 244, 912–916 (2014). doi:10.1016/j.amc.2014.07.022
Guo, B.-N., Qi, F.: A completely monotonic function involving the tri-gamma function and with degree one. Appl. Math. Comput. 218(19), 9890–9897 (2012). doi:10.1016/j.amc.2012.03.075
Guo, B.-N., Qi, F.: A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function. Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 72(2), 21–30 (2010)
Guo, B.-N., Qi, F.: On the degree of the weighted geometric mean as a complete Bernstein function. Afr. Mat. 26(7), 1253–1262 (2015). doi:10.1007/s13370-014-0279-2
Guo, S., Qi, F.: A class of completely monotonic functions related to the remainder of Binet’s formula with applications. Tamsui Oxf. J. Math. Sci. 25(1), 9–14 (2009)
Koumandos, S., Pedersen, H.L.: Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler’s gamma function. J. Math. Anal. Appl. 355(1), 33–40 (2009). doi:10.1016/j.jmaa.2009.01.042
Magnus, W., Oberhettinger, F., Soni, R.P.: Formulas and Theorems for the Special Functions of Mathematical Physics. Springer, Berlin (1966). doi:10.1137/1009129
Mitrinović, D.S., Pečarić, J.E., Fink, A.M.: Classical and New Inequalities in Analysis. Kluwer Academic Publishers, Dordrecht (1993). doi:10.1007/978-94-017-1043-5
Mortici, C.: On the monotonicity and convexity of the remainder of the Stirling formula. Appl. Math. Lett. 24(6), 869–871 (2011). doi:10.1016/j.aml.2010.12.034
Qi, F.: Completely monotonic degree of a function involving the tri- and tetra-gamma functions. arXiv:1301.0154
Qi, F.: Integral representations and complete monotonicity related to the remainder of Burnside’s formula for the gamma function. J. Comput. Appl. Math. 268, 155–167 (2014). doi:10.1016/j.cam.2014.03.004
Qi, F.: Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions. Math. Inequal. Appl. 18(2), 493–518 (2015). doi:10.7153/mia-18-37
Qi, F., Chen, C.-P.: A complete monotonicity property of the gamma function. J. Math. Anal. Appl. 296, 603–607 (2004). doi:10.1016/j.jmaa.2004.04.026
Qi, F., Guo, B.-N.: Complete monotonicities of functions involving the gamma and digamma functions. RGMIA Res. Rep. Coll. 7(1) (2004), 63–72, Art. 8. Available online at http://rgmia.org/v7n1.php
Qi, F., Guo, B.-N.: Integral representations and complete monotonicity of remainders of the Binet and Stirling formulas for the gamma function. ResearchGate Technical Report. doi:10.13140/2.1.2733.3928
Qi, F., Guo, B.-N.: Some properties of extended remainder of Binet’s first formula for logarithm of gamma function. Math. Slovaca 60(4), 461–470 (2010). doi:10.2478/s12175-010-0025-7
Qi, F., Guo, S., Guo, B.-N.: Complete monotonicity of some functions involving polygamma functions. J. Comput. Appl. Math. 233(9), 2149–2160 (2010). doi:10.1016/j.cam.2009.09.044
Qi, F., Li, W.-H.: A logarithmically completely monotonic function involving the ratio of gamma functions. J. Appl. Anal. Comput. 5(4), 626–634 (2015). doi:10.11948/2015049
Qi, F., Luo, Q.-M., Guo, B.-N.: Complete monotonicity of a function involving the divided difference of digamma functions. Sci. China Math. 56(11), 2315–2325 (2013). doi:10.1007/s11425-012-4562-0
Qi, F., Luo, Q.-M., Guo, B.-N.: The function \((b^x-a^x)/x\): Ratio’s properties. In: Milovanović, G.V., Rassias, M.T. (eds.) Analytic Number Theory, Approximation Theory, and Special Functions, pp. 485–494. Springer, Berlin (2014). doi:10.1007/978-1-4939-0258-3_16
Qi, F., Wang, S.-H.: Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions. Glob. J. Math. Anal. 2(3), 91–97 (2014). doi:10.14419/gjma.v2i3.2919
Qi, F., Wei, C.-F., Guo, B.-N.: Complete monotonicity of a function involving the ratio of gamma functions and applications. Banach J. Math. Anal. 6(1), 35–44 (2012). doi:10.15352/bjma/1337014663
Schilling, R.L., Song, R., Vondraček, Z.: Bernstein Functions—Theory and Applications, de Gruyter Studies in Mathematics, vol. 37, 2nd edn. Walter de Gruyter, Berlin (2012). doi:10.1515/9783110269338
Shi, X.-Q., Liu, F.-S., Hu, M.-H.: A new asymptotic series for the gamma function. J. Comput. Appl. Math. 195(1–2), 134–154 (2006). doi:10.1016/j.cam.2005.03.081
Trimble, S.Y., Wells, J., Wright, F.T.: Superadditive functions and a statistical application. SIAM J. Math. Anal. 20(5), 1255–1259 (1989). doi:10.1137/0520082
Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. Reprint of the 4th edn. Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. doi:10.1017/CBO9780511608759 (1927)
Widder, D.V.: The Laplace Transform, Princeton Mathematical Series, vol. 6. Princeton University Press, Princeton (1941)
Zhang, S.-Q., Guo, B.-N., Qi, F.: A concise proof for properties of three functions involving the exponential function. Appl. Math. E Notes 9, 177–183 (2009)
Acknowledgments
The authors thank Dr. Songbai Guo in China and Professor Cristinel Mortici in Romania for their helpfully commenting on, carefully correcting to, and patiently checking up the original version of this paper.
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Qi, F., Guo, BN. Integral representations and complete monotonicity of remainders of the Binet and Stirling formulas for the gamma function. RACSAM 111, 425–434 (2017). https://doi.org/10.1007/s13398-016-0302-6
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DOI: https://doi.org/10.1007/s13398-016-0302-6
Keywords
- Remainder
- Binet formula
- Stirling formula
- Gamma function
- Complete monotonicity
- Integral representation
- Limit