where is the Riemann
zeta function, which was subsequently rigorously proven true (Borwein and Borwein
1995). Sums involving
can be re-expressed in terms of sums the form via
(4)
(5)
(6)
and
(7)
where is defined below.
Bailey et al. (1994) subsequently considered sums of
the forms
for , given as a challenge problem by Borwein
and Bailey (2003, pp. 24-25) and discussed in Bailey et al. (2006a, p. 39;
Bailey et al. 2006b),
(44)
(45)
(46)
for , and
(47)
(48)
for , where is a polylogarithm, and
is the Riemann
zeta function (Bailey and Plouffe 1997, Bailey et al. 1994). Of these,
only (P. Simone, pers. comm.,
Aug. 30, 2004), ,
and the identities for , and have been rigorously established.
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and Moll, V. H. Experimental
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Notebooks: Part I. New York: Springer-Verlag, 1985.Borwein,
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Some Intriguing Sums Involving ." Proc. Amer. Math. Soc.123, 111-118,
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and Some Series Related to ."
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Values of Multidimensional Polylogarithms." Trans. Amer. Math. Soc.353,
907-941, 2001.Boyadzhiev, K. N. "Evaluation of Euler-Zagier
Sums." Int. J. Math. Math. Sci.27, 407-412, 2001. http://www2.onu.edu/~kboyadzh/e-zagier.pdf.Boyadzhiev,
K. N. "Consecutive Evaluation of Euler Sums." Int. J. Math. Math.
Sci.29, 555-561, 2002. http://www2.onu.edu/~kboyadzh/euler-c(1).pdf.Broadhurst,
D. J. "On the Enumeration of Irreducible -Fold Euler Sums and Their Roles in Knot Theory and Field Theory."
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