Advertisement

The Ramanujan Journal

, Volume 36, Issue 1–2, pp 43–68 | Cite as

On Eulerian log-gamma integrals and Tornheim–Witten zeta functions

  • David H. BaileyEmail author
  • David Borwein
  • Jonathan M. Borwein
Research Paper

Abstract

Stimulated by earlier work by Moll and his coworkers (Amdeberhan et al., Proc. Am. Math. Soc., 139(2):535–545, 2010), we evaluate various basic log Gamma integrals in terms of partial derivatives of Tornheim–Witten zeta functions and their extensions arising from evaluations of Fourier series. In particular, we fully evaluate
$$\mathcal{LG}_n=\int_0^1 \log^n\varGamma(x) \,\mathrm{d}x $$
for 1≤n≤4 and make some comments regarding the general case. The subsidiary computational challenges are substantial, interesting and significant in their own right.

Keywords

Gamma function Log gamma function Riemann zeta function Tornheim–Weitten zeta function Integration 

Mathematics Subject Classification

11M35 11M32 33B15 

Notes

Acknowledgements

Thanks are due to Victor Moll for suggesting we revisit this topic and to Richard Crandall for his significant help with computation of ω’s partial derivatives and for his insightful solutions to various other of our computational problems.

References

  1. 1.
    Amdeberhan, T., Coffey, M., Espinosa, O., Koutschan, C., Manna, D., Moll, V.: Integrals of powers of loggamma. Proc. Am. Math. Soc. 139(2), 535–545 (2010) CrossRefMathSciNetGoogle Scholar
  2. 2.
    Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999) CrossRefzbMATHGoogle Scholar
  3. 3.
    Bailey, D.H., Borwein, J.M., Crandall, R.E.: Computation and theory of extended Mordell–Tornheim–Witten sums (2012). Preprint Google Scholar
  4. 4.
    Borwein, J.: Hilbert inequalities and Witten zeta-functions. Proc. Am. Math. Soc. 115(2), 125–137 (2008) zbMATHGoogle Scholar
  5. 5.
    Borwein, J., Bailey, D., Girgensohn, R., Luke, R., Moll, V.: Experimental Mathematics in Action. AK Peters, Wellesley (2007) zbMATHGoogle Scholar
  6. 6.
    Borwein, J.M., Bradley, D.M.: Thirty two Goldbach variations. Int. J. Number Theory 2(1), 65–103 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Borwein, J.M., Bradley, D.M., Broadhurst, D.J., Lisoněk, P.: Special values of multidimensional polylogarithms. Trans. Am. Math. Soc. 353(3), 907–941 (2001) CrossRefzbMATHGoogle Scholar
  8. 8.
    Borwein, J.M., Straub, A.: Special values of generalized log-sine integrals. In: Proceedings of ISSAC 2011 (International Symposium on Symbolic and Algebraic Computation) (2011) Google Scholar
  9. 9.
    Borwein, J.M., Zucker, I.J., Boersma, J.: The evaluation of character Euler double sums. Ramanujan J. 15(3), 377–405 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Crandall, R.: Unified algorithms for polylogarithm, L-series, and zeta variants (2012). Preprint Google Scholar
  11. 11.
    Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.: Higher Transcendental Functions vol. 1, 2nd edn. Kreiger, Huntington (1981) Google Scholar
  12. 12.
    Espinosa, O., Moll, V.: The evaluation of Tornheim double sums. Part I. J. Number Theory 116, 200–229 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Espinosa, O., Moll, V.: The evaluation of Tornheim double sums. Part II. J. Number Theory 22, 55–99 (2010) zbMATHMathSciNetGoogle Scholar
  14. 14.
    Matsumoto, K., Tsumara, H.: A new method of producing functional relations among multiple zeta-functions. Q. J. Math. 59, 55–83 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Nielsen, N.: Handbuch der Theorie der Gammafunction. Teubner, Leipzig (1906) Google Scholar
  16. 16.
    Stromberg, K.R.: An Introduction to Classical Real Analysis. Wadsworth, Belmont (1981) zbMATHGoogle Scholar
  17. 17.
    Wood, D.: The computation of polylogarithms. Technical report 15-92. Canterbury, UK: University of Kent Computing Laboratory (1992). Retrieved 2005-11-01 Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • David H. Bailey
    • 1
    Email author
  • David Borwein
    • 2
  • Jonathan M. Borwein
    • 3
    • 4
  1. 1.Lawrence Berkeley National LaboratoryBerkeleyUSA
  2. 2.Department of MathematicsUniversity of Western OntarioLondonCanada
  3. 3.Centre for Computer-assisted Research Mathematics and its Applications (CARMA), School of Mathematical and Physical SciencesUniversity of NewcastleCallaghanAustralia
  4. 4.King Abdul-Aziz UniversityJeddahSaudia Arabia

Personalised recommendations