Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-30T04:01:06.486Z Has data issue: false hasContentIssue false
  • English
  • Français

Certain functional relations for the double harmonic series related to the double Euler numbers

Part of: Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  09 April 2009

Hirofumi Tsumura
Hirofumi Tsumura
Affiliation:
Department of Mathematics, Tokyo Metropolitan University, 1-1, Minami-Osawa, Hachioji, Tokyo 192-0397, Japan, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we give certain analytic functional relations for the double harmonic series related to the double Euler numbers. These can be regarded as continuous generalizations of the known discrete relations obtained by the author recently.

Keywords

double Euler numbersdouble harmonic seriesMordell-Tornheim double zeta functionsRiemann zeta functionuniformly convergent series.

MSC classification

Secondary: 40B05: Multiple sequences and series (should also be assigned at least one other classification number in this section) 11M41: Other Dirichlet series and zeta functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 79 , Issue 3 , December 2005 , pp. 319 - 333
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Dilcher, K., Zeros of Bernoulli, generalized Bernoulli and Euler polynomials, Mem. Amer. Math. Soc. 386 (Amer. Math. Soc., Providence, RI, 1988).Google Scholar
[2]Huard, J. G., Williams, K. S. and Nan-Yue, Z., ‘On Tornheim's double series’, Acta Arith. 75 (1996), 105–117.CrossRefGoogle Scholar
[3]Matsumoto, K., ‘On Mordell-Tornheim and other multiple zeta-functions’, in: Proceedings of the Session in analytic number theory and Diophantine equations. Bonn, January-June 2002 (eds. Heath-Brown, D. R. and Moroz, B. Z.), Bonner Mathematische Schriften 360 (Bonn, 2003) pp. 17.Google Scholar
[4]Mordell, L. J., ‘On the evaluation of some multiple series’, J. London Math. Soc. 33 (1958), 368–371.CrossRefGoogle Scholar
[5]Subbarao, M. V. and Sitaramachandrarao, R., ‘On some infinite series of L. J. Mordell and their analogues’, Pacific J. Math. 119 (1985), 245–255.CrossRefGoogle Scholar
[6]Tornheim, L., ‘Harmonic double series’, Amer. J. Math. 72 (1950), 303–314.CrossRefGoogle Scholar
[7]Tsumura, H., ‘On functional relations between the Mordell-Tornheim double zeta functions and the Riemann zeta function’, preprint.Google Scholar
[8]Tsumura, H., ‘On some combinatorial relations for Tornheim's double series’, Acta Arith. 105 (2002), 239–252.CrossRefGoogle Scholar
[9]Tsumura, H., ‘On alternating analogues of Tornheim's double series’, Proc. Amer. Math. Soc. 131 (2003), 3633–3641.CrossRefGoogle Scholar
[10]Tsumura, H., ‘Multiple harmonic series related to multiple Euler numbers’, J. Number Theory 106 (2004), 155–168.CrossRefGoogle Scholar
[11]Washington, L. C., Introduction to the cyclotomic fields, 2nd edition (Springer, New York, 1997).CrossRefGoogle Scholar