Advertisement

Monatshefte für Mathematik

, Volume 163, Issue 1, pp 51–69 | Cite as

Absolutely monotonic functions related to Euler’s gamma function and Barnes’ double and triple gamma function

  • Stamatis Koumandos
  • Henrik L. PedersenEmail author
Article

Abstract

It is shown that the remainders in asymptotic expansions of the logarithm of Barnes double and triple gamma function and Euler’s gamma function are Laplace transformations of positive multiples of absolutely monotonic functions. Applications concerning positivity of sums involving Bernoulli numbers are given.

Keywords

Double gamma function Triple gamma function Asymptotic expansion Absolute monotonicity 

Mathematics Subject Classification (2000)

Primary 33B15 Secondary 41A60 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alzer H.: On some inequalities for the gamma and psi functions. Math. Comp. 66(217), 373–389 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Barnes E.W.: The theory of the gamma function. Messenger Math. 29, 64–128 (1900)Google Scholar
  3. 3.
    Barnes E.W.: The theory of the G-function. Quart. J. Math. 31, 264–314 (1899)Google Scholar
  4. 4.
    Barnes E.W.: The genesis of the double gamma functions. Proc. Lond. Math. Soc. 31, 358–381 (1901)CrossRefGoogle Scholar
  5. 5.
    Barnes E.W.: The theory of the double gamma function. Philos. Trans. Roy. Soc. Lond. 196, 265–388 (1901)CrossRefGoogle Scholar
  6. 6.
    Barnes E.W.: The theory of the multiple gamma function. Trans. Camb. Philos. Soc. 19, 374–425 (1904)Google Scholar
  7. 7.
    Gradshteyn I.S., Ryzhik I.M.: Tables of Integrals, Series, and Products, 4th edn. Academic Press, New York, London (1965)Google Scholar
  8. 8.
    Koumandos S.: Remarks on some completely monotonic functions. J. Math. Anal. Appl. 324, 1458–1461 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Koumandos S.: On Ruijsenaars’ asymptotic expansion of the logarithm of the double gamma function. J. Math. Anal. Appl. 341, 1125–1132 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Koumandos S., Pedersen H.L.: On the asymptotic expansion of the logarithm of Barnes triple gamma function. Math. Scand. 105, 287–306 (2009)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Ruijsenaars S.N.M.: On Barnes’ multiple zeta and gamma functions. Adv. Math. 156, 107–132 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Pedersen H.L.: The remainder in Ruijsenaars’ asymptotic expansion of Barnes double gamma function. Mediterr. J. Math. 4, 419–433 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Temme N.M.: Special Functions: An Introduction to Classical Functions of Mathematical Physics. Wiley, New York (1996)zbMATHGoogle Scholar
  15. 15.
    Widder D.V.: The Laplace Transform. Princeton University Press, Princeton (1941)Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsThe University of CyprusNicosiaCyprus
  2. 2.Department of Basic Sciences and Environment, Mathematics and Computer Science, Faculty of Life SciencesUniversity of CopenhagenFrederiksberg CDenmark

Personalised recommendations