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

Published online by Cambridge University Press:  09 April 2009

Hirofumi Tsumura
Affiliation:
Department of Mathematics, Tokyo Metropolitan University, 1-1, Minami-Osawa, Hachioji, Tokyo 192-0397, Japan, e-mail: tsumura@comp.metro-u.ac.jp
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Abstract

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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.

Type
Research Article
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), 105117.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), 368371.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), 245255.CrossRefGoogle Scholar
[6]Tornheim, L., ‘Harmonic double series’, Amer. J. Math. 72 (1950), 303314.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), 239252.CrossRefGoogle Scholar
[9]Tsumura, H., ‘On alternating analogues of Tornheim's double series’, Proc. Amer. Math. Soc. 131 (2003), 36333641.CrossRefGoogle Scholar
[10]Tsumura, H., ‘Multiple harmonic series related to multiple Euler numbers’, J. Number Theory 106 (2004), 155168.CrossRefGoogle Scholar
[11]Washington, L. C., Introduction to the cyclotomic fields, 2nd edition (Springer, New York, 1997).CrossRefGoogle Scholar