Evaluation of Tornheim’s type of double series. (English) Zbl 1434.11185

Kadota, Shin-Ya; Okamoto, Takuya; Tasaka, Koji

Ill. J. Math. 61, No. 1-2, 171-186 (2017).

Summary: We examine values of certain Tornheim’s type of double series with odd weight. As a result, an affirmative answer to a conjecture about the parity theorem for the zeta function of the root system of the exceptional Lie algebra \(G_2\), proposed by Y. Komori et al. [Glasg. Math. J. 53, No. 1, 185–206 (2011; Zbl 1255.11042)], is given.

MSC:

11M32	Multiple Dirichlet series and zeta functions and multizeta values
17B22	Root systems

Keywords:

double series; Lie algebra

Citations:

Zbl 1255.11042

References:

[1]	T. Arakawa, T. Ibukiyama and M. Kaneko, Bernoulli numbers and zeta functions, Springer Monographs in Mathematics, Springer, Tokyo, 2014. · Zbl 1312.11015
[2]	J. G. Huard, K. S. Williams and N. Y. Zhang, On Tornheim’s double series, Acta Arith. 75 (1996), no. 2, 105-117. · Zbl 0858.40008
[3]	K. Ihara, M. Kaneko and D. Zagier, Derivation and double shuffle relations for multiple zeta values, Compos. Math. 142 (2006), no. 2, 307-338. · Zbl 1186.11053
[4]	Y. Komori, K. Matsumoto and H. Tsumura, On Witten multiple zeta-functions associated with semi-simple Lie algebras IV, Glasg. Math. J. 53 (2011), no. 1, 185-206. · Zbl 1255.11042 · doi:10.1017/S0017089510000613
[5]	Y. Komori, K. Matsumoto and H. Tsumura, On Witten multiple zeta-function associated with semi-simple Lie algebras V, Glasg. Math. J. 57 (2015), no. 1, 107-130. · Zbl 1318.11114 · doi:10.1017/S0017089514000160
[6]	T. Nakamura, A functional relation for the Tornheim double zeta function, Acta Arith. 125 (2006), no. 3, 257-263. · Zbl 1153.11047
[7]	T. Nakamura, A simple proof of the functional relation for the Lerch type Tornheim double zeta function, Tokyo J. Math. 35 (2012), no. 2, 333-337. · Zbl 1276.11143
[8]	T. Okamoto, Multiple zeta values related with the zeta-function of the root system of type \(A_2, B_2\) and \(G_2\), Comment. Math. Univ. St. Pauli 61 (2012), no. 1, 9-27. · Zbl 1321.11091
[9]	T. Okamoto, On alternating analogues of the Mordell-Tornheim triple zeta values, J. Ramanujan Math. Soc. 28 (2013), no. 2, 247-269. · Zbl 1368.11101
[10]	E. Panzer, The parity theorem for multiple polylogarithms, J. Number Theory 172 (2017), 93-113. · Zbl 1419.11109 · doi:10.1016/j.jnt.2016.08.004
[11]	M. V. Subbarao and R. Sitaramachandra, On some infinite series of L. J. Mordell and their analogues, Pacific J. Math. 119 (1985), no. 1, 245-255. · Zbl 0573.10026
[12]	L. Tornheim, Harmonic double series, Amer. J. Math. 72 (1950), 303-314. · Zbl 0036.17203 · doi:10.2307/2372034
[13]	H. Tsumura, On alternating analogues of Tornheim’s double series, Proc. Amer. Math. Soc. 131 (2003), no. 12, 3633-3641. · Zbl 1045.11057
[14]	H. Tsumura, Evaluation formulas for Tornheim’s type of alternating double series, Math. Comp. 73 (2004), no. 245, 251-258. · Zbl 1094.11034
[15]	H. Tsumura, Combinatorial relations for Euler-Zagier sums, Acta Arith. 111 (2004), no. 1, 27-42. · Zbl 1153.11327
[16]	H. Tsumura, On Mordell-Tornheim zeta values, Proc. Amer. Math. Soc. 133 (2005), 2387-2393. · Zbl 1059.40003
[17]	H. Tsumura, On alternating analogues of Tornheim’s double series. II, Ramanujan J. 18 (2009), no. 1, 81-90. · Zbl 1192.11059
[18]	J. Zhao, Multi-polylogs at twelfth roots of unity and special values of Witten multiple zeta function attached to the exceptional Lie algebra \(\mathfrak{g}_2 \), J. Algebra Appl. 9 (2010), no. 2, 327-337. · Zbl 1197.11108
[19]	X. Zhou, T. Cai and D. M. Bradley, Signed \(q\)-analogs of Tornheim’s double series, Proc. Amer. Math. Soc. 136 (2008), no. 8, 2689-2698. · Zbl 1204.11150