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I am studying Multizeta values at the moment and I found that at weight 5, the basis is given by ζ(5) and ζ(3)ζ(2) in the literature. Solving all shuffle and stuffle relations using mathematica I, however, get as a basis ζ(5) and ζ(4,1).

Are there any further relations which relate ζ(3)ζ(2) and ζ(4,1) so that they are secretely the same thing?

P.s.: I've also asked this question on math.stackexchange.

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The relation is $\zeta(4,1)=2\zeta(5)-\zeta(2)\zeta(3)$. It can be found, for example, in http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.pjm/1102636166&page=record (Multiple harmonic series by Michael E. Hoffman, p.281). Note that $\zeta(4,1)=A(4,1)$ and $S(4,1)=A(4,1)+A(5)$ in Hoffman's notations. Hoffman attributes the identity (in fact more general identity) to Euler.

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  • $\begingroup$ Thanks for your answer. THis was what I was looking for. One more question though: can this be derived from shuffle and stuffle relations? It doesn't look like it's a consequence of Shuffles or Stuffle relations? (I'll check out your reference later today) $\endgroup$
    – A friendly helper
    Commented Mar 13, 2014 at 20:40

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