Abstract
We present a summary of the series representations of the remainders in the expansions in ascending powers of t of \({2/(e^t+1)}\), sech t and coth t and establish simple bounds for these remainders when \({t > 0}\). Several applications of these expansions are given which enable us to deduce some inequalities and completely monotonic functions associated with the ratio of two gamma functions. In addition, we derive a (presumably new) quadratic recurrence relation for the Bernoulli numbers B n .
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz, M., Stegun, I.A. (eds.) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. In: Applied Mathematics Series 55, Ninth printing. National Bureau of Standards, Washington, D.C. (1972)
Alzer H.: On some inequalities for the gamma and psi functions. Math. Comput. 66, 373–389 (1997)
Beckmann, P.: A History of Pi. St. Martin’s Press, New York (1971)
Berggren, L., Borwein, J., Borwein, P. (eds.) Pi: a source book, 2nd edn. Springer, New York (2000)
Berndt B.C.: Ramanujan’s Notebooks, Part V. Springer, Berlin (1998)
Chen C.-P.: Inequalities and completely monotonic functions associated with the ratios of functions resulting from the gamma function. Appl. Math. Comput. 259, 790–799 (2015)
Chen C.-P., Qi F.: Completely monotonic function associated with the gamma functions and proof of Wallis’ inequality. Tamkang J. Math. 36, 303–307 (2005)
Chen C.-P., Qi F.: The best bounds in Wallis’ inequality. Proc. Am. Math. Soc. 133, 397–401 (2005)
Chen C.-P., Paris R.B.: Inequalities, asymptotic expansions and completely monotonic functions related to the gamma function. Appl. Math. Comput. 250, 514–529 (2015)
Dunham, W.: Journey Through Genius. In: The Great Theorems of Mathematics, Penguin (1990)
Ferreira C., López J.L.: An asymptotic expansion of the double gamma function. J. Approx. Theory 111, 298–314 (2001)
Koumandos S.: Remarks on some completely monotonic functions. J. Math. Anal. Appl. 324, 1458–1461 (2006)
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, 33–40 (2009)
Koumandos S.: Remarks on a paper by Chao-Ping Chen and Feng Qi. Proc. Am. Math. Soc. 134, 1365–1367 (2006)
Koumandos S.: On completely monotonic and related functions. Mathematics Without Boundaries, pp. 285–321. Springer, New York (2014)
Lampret V.: Wallis sequence estimated through the Euler–Maclaurin formula: even from the Wallis product \({\pi}\) could be computed fairly accurately. Austr. Math. Soc. Gaz. 31, 328–339 (2004)
Lampret V.: An asymptotic approximation of Wallis’ sequence. Cent. Eur. J. Math. 10, 775–787 (2012)
Matiyasevich, Y.: Identities with Bernoulli numbers. Identity #0202 (1997). http://logic.pdmi.ras.ru/~yumat/Journal/Bernoulli/bernoulli.htm. Accessed 23 Apr 2015
Mortici C.: New approximation formulas for evaluating the ratio of gamma functions. Math. Comput. Model. 52, 425–433 (2010)
Mortici C., Cristea V.G., Lu D.: Completely monotonic functions and inequalities associated to some ratio of gamma function. Appl. Math. Comput. 240, 168–174 (2014)
Muldoon M.E.: Some monotonicity properties and characterizations of the gamma function. Aequ. Math. 18, 54–63 (1978)
Olver F.W.J., Lozier D.W., Boisvert R.F., Clarks, C.W. (eds.) NIST Handbook of Mathematical Functions. Cambridge University Press, New York (2010)
Pedersen H.L.: On the remainder in an asymptotic expansion of the double gamma function. Mediterr. J. Math. 2, 171–178 (2005)
Pilehrood K.H., Pilehrood T.H.: Arithmetical properties of some series with logarithmic coefficients. Math. Z. 255, 117–131 (2007)
Pólya G., Szegö G.: Problems and Theorems in Analysis, vol. I, II. Springer, Berlin (1972)
Sasvári Z.: Inequalities for binomial coefficients. J. Math. Anal. Appl. 236, 223–226 (1999)
Sofo A.: Some representations of \({\pi}\), Austral. Math. Soc. Gaz. 31, 184–189 (2004)
Sofo, A.: \({\pi}\) and some other constants. J. Inequal. Pure Appl. Math. 6(5) (article 138) (electronic) (2005)
Sondow J.: Double integrals for Euler’s constant and \({\ln(4/\pi)}\) and an analog of Hadjicostas’s formula. Am. Math. Monthly 112, 61–65 (2005)
Sondow, J.: New Vacca-type rational series for Euler’s constant and its alternating analog \({\ln(4/\pi)}\) In: Chudnovsky, D., Chudnovsky, G. (eds.) Additive Number Theory. Festschrift in Honor of the Sixtieth Birthday of Melvyn B. Nathanson, pp. 331–340. Springer, Berlin (2010)
Sondow J., Hadjicostas P.: The generalized-Euler-constant function \({\gamma(z)}\) and a generalization of Somos’s quadratic recurrence constant. J. Math. Anal. Appl. 332, 292–314 (2007)
Srivastava, H.M., Choi J.: Series Associated with the Zeta and Related Functions. Kluwer Academic Publishers, Dordrecht (2001)
Temme N.M.: Special Functions: An Introduction to the Classical Functions of Mathematical Physics. Wiley, New York (1996)
Wang Z.X., Guo D.R.: Special Functions. World Scientific, Singapore (1989)
Weisstein, E.W.: Bernoulli Numbers. From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/BernoulliNumber.html. Accessed 10 Apr 2015
Whittaker E.T., Watson G.N.: A Course of Modern Analysis, 4th edn. Cambridge University Press, Cambridge (1966)
Wikipedia contributors, Bernoulli numbers, Wikipedia, the free encyclopedia. https://en.wikipedia.org/wiki/Bernoullinumbers. Accessed 10 Apr 2015
Xu Y., Han X.: Complete monotonicity properties for the gamma function and Barnes G-function. Sci. Magna 5, 47–51 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, CP., Paris, R.B. Series Representations of the Remainders in the Expansions for Certain Functions with Applications. Results Math 71, 1443–1457 (2017). https://doi.org/10.1007/s00025-016-0612-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-016-0612-1