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

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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.

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References

  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)

    Article  MathSciNet  Google Scholar 

  2. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)

    Book  MATH  Google Scholar 

  3. Bailey, D.H., Borwein, J.M., Crandall, R.E.: Computation and theory of extended Mordell–Tornheim–Witten sums (2012). Preprint

  4. Borwein, J.: Hilbert inequalities and Witten zeta-functions. Proc. Am. Math. Soc. 115(2), 125–137 (2008)

    MATH  Google Scholar 

  5. Borwein, J., Bailey, D., Girgensohn, R., Luke, R., Moll, V.: Experimental Mathematics in Action. AK Peters, Wellesley (2007)

    MATH  Google Scholar 

  6. Borwein, J.M., Bradley, D.M.: Thirty two Goldbach variations. Int. J. Number Theory 2(1), 65–103 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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. Borwein, J.M., Zucker, I.J., Boersma, J.: The evaluation of character Euler double sums. Ramanujan J. 15(3), 377–405 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Crandall, R.: Unified algorithms for polylogarithm, L-series, and zeta variants (2012). Preprint

  11. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.: Higher Transcendental Functions vol. 1, 2nd edn. Kreiger, Huntington (1981)

    Google Scholar 

  12. Espinosa, O., Moll, V.: The evaluation of Tornheim double sums. Part I. J. Number Theory 116, 200–229 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Espinosa, O., Moll, V.: The evaluation of Tornheim double sums. Part II. J. Number Theory 22, 55–99 (2010)

    MATH  MathSciNet  Google Scholar 

  14. Matsumoto, K., Tsumara, H.: A new method of producing functional relations among multiple zeta-functions. Q. J. Math. 59, 55–83 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Nielsen, N.: Handbuch der Theorie der Gammafunction. Teubner, Leipzig (1906)

    Google Scholar 

  16. Stromberg, K.R.: An Introduction to Classical Real Analysis. Wadsworth, Belmont (1981)

    MATH  Google Scholar 

  17. Wood, D.: The computation of polylogarithms. Technical report 15-92. Canterbury, UK: University of Kent Computing Laboratory (1992). Retrieved 2005-11-01

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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.

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Authors and Affiliations

  1. Lawrence Berkeley National Laboratory, Berkeley, CA, 94720, USA

    David H. Bailey

  2. Department of Mathematics, University of Western Ontario, London, ON, Canada

    David Borwein

  3. Centre for Computer-assisted Research Mathematics and its Applications (CARMA), School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW, 2308, Australia

    Jonathan M. Borwein

  4. King Abdul-Aziz University, Jeddah, Saudia Arabia

    Jonathan M. Borwein

Authors
  1. David H. Bailey
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  2. David Borwein
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  3. Jonathan M. Borwein
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Corresponding author

Correspondence to David H. Bailey.

Additional information

Dedicated to the memory of Basil Gordon

D.H. Bailey was supported in part by the Director, Office of Computational and Technology Research, Division of Mathematical, Information, and Computational Sciences of the US Department of Energy, under contract number DE-AC02-05CH11231.

J.M. Borwein was supported in part by the Australian Research Council and the University of Newcastle.

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Bailey, D.H., Borwein, D. & Borwein, J.M. On Eulerian log-gamma integrals and Tornheim–Witten zeta functions. Ramanujan J 36, 43–68 (2015). https://doi.org/10.1007/s11139-012-9427-1

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  • DOI: https://doi.org/10.1007/s11139-012-9427-1

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