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

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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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.

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

1. David H. Bailey
2. David Borwein
3. Jonathan M. Borwein

Correspondence to David H. Bailey.

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.

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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• Received: 21 February 2012

• Accepted: 22 July 2012

• Published: 27 February 2013

• Issue Date: February 2015

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

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