# zbMATH — the first resource for mathematics

A generalization of the duality and sum formulas on the multiple zeta values. (English) Zbl 0920.11063
The author generalizes a duality theorem of D. Zagier [Prog. Math. 120, 497–512 (1994; Zbl 0822.11001)] on multiple zeta values from which several results of M. Hoffman [Pac. J. Math. 152, 275–290 (1992; Zbl 0763.11037)] and others are deduced as special cases. Another application is the evaluation of the integral $\xi_k(s)= {1\over\Gamma (s)} \int^\infty_0{t^{s-1} \over e^t-1} Li_k(1-e^{-t})\, dt$ for positive integer values of $$s$$, where $$Li_k(z)$$ denotes the $$k$$th polylogarithm $$\sum^\infty_{m=0} m^{-k}z^m$$.

##### MSC:
 11M32 Multiple Dirichlet series and zeta functions and multizeta values
Full Text:
##### References:
 [1] T. Arakawa, M. Kaneko, Multiple zeta values, poly-Bernoulli numbers, and related zeta functions, Nagoya Math. J. · Zbl 0932.11055 [2] T. Arakawa, M. Kaneko, Multiple zeta values, poly-Bernoulli numbers, and related zeta functions, Proceedings, Symposium in Tsudajyuku Univ, 1997, 2, 133, 144 · Zbl 0932.11055 [3] Borwein, D.; Borwein, J.M.; Girgensohn, R., Explicit evaluation of Euler sums, Proc. Edinburgh math. soc., 38, 277-294, (1995) · Zbl 0819.40003 [4] Hoffman, M., Multiple harmonic series, Pacific J. math., 152, 275-290, (1992) · Zbl 0763.11037 [5] Huard, J.G.; Williams, K.S.; Nan-Yue, Zhang, On Tornheim’s double series, Acta arith., 75, 105-117, (1996) · Zbl 0858.40008 [6] Kaneko, M., Poly-Bernoulli numbers, J. théor. nombres Bordeaux, 9, 221-228, (1997) · Zbl 0887.11011 [7] Le, T.Q.T.; Murakami, J., Kontsevich’s integral for the homfly polynomial and relations between values of multiple zeta functions, Topology appl., 62, 193-206, (1995) · Zbl 0839.57007 [8] Lewin, L., Polylogarithms and associated functions, (1980), Tata Bombay [9] Zagier, D., Values of zeta functions and their applications, in ECM volume, Progr. math., 120, 497-512, (1994) · Zbl 0822.11001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.