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Multiple zeta values

Table of contents: Definitions - Sum representations - Specific values - Families of closed forms - Relations

Definitions

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Symbol: MultiZetaValue ζ ⁣(s1,,sk)\zeta\!\left({s}_{1}, \ldots, {s}_{k}\right) Multiple zeta value (MZV)

Sum representations

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ζ ⁣(s1,s2,,sk)=nZkn1>n2>>nk>0i=1k1nisi\zeta\!\left({s}_{1}, {s}_{2}, \ldots, {s}_{k}\right) = \sum_{\textstyle{n \in {\mathbb{Z}}^{k} \atop {n}_{1} > {n}_{2} > \ldots > {n}_{k} > 0}} \prod_{i=1}^{k} \frac{1}{{n}_{i}^{{s}_{i}}}

Specific values

345c26
ζ ⁣(2,1)=n=1Hn(n+1)2=ζ ⁣(3)\zeta\!\left(2, 1\right) = \sum_{n=1}^{\infty} \frac{H_{n}}{{\left(n + 1\right)}^{2}} = \zeta\!\left(3\right)
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ζ ⁣(2,2)=34ζ ⁣(4)\zeta\!\left(2, 2\right) = \frac{3}{4} \zeta\!\left(4\right)
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ζ ⁣(3,2)=3ζ ⁣(2)ζ ⁣(3)112ζ ⁣(5)\zeta\!\left(3, 2\right) = 3 \zeta\!\left(2\right) \zeta\!\left(3\right) - \frac{11}{2} \zeta\!\left(5\right)
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ζ ⁣(4,2)=(ζ ⁣(3))243ζ ⁣(6)\zeta\!\left(4, 2\right) = {\left(\zeta\!\left(3\right)\right)}^{2} - \frac{4}{3} \zeta\!\left(6\right)
856317
ζ ⁣(2,3)=92ζ ⁣(5)2ζ ⁣(2)ζ ⁣(3)\zeta\!\left(2, 3\right) = \frac{9}{2} \zeta\!\left(5\right) - 2 \zeta\!\left(2\right) \zeta\!\left(3\right)
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ζ ⁣(3,3)=12((ζ ⁣(3))2ζ ⁣(6))\zeta\!\left(3, 3\right) = \frac{1}{2} \left({\left(\zeta\!\left(3\right)\right)}^{2} - \zeta\!\left(6\right)\right)

Families of closed forms

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ζ ⁣(s,s)=12((ζ ⁣(s))2ζ ⁣(2s))\zeta\!\left(s, s\right) = \frac{1}{2} \left({\left(\zeta\!\left(s\right)\right)}^{2} - \zeta\!\left(2 s\right)\right)
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ζ ⁣(2,,2n times)=π2n(2n+1)!\zeta\!\left(\underbrace{2, \ldots, 2}_{n \text{ times}}\right) = \frac{{\pi}^{2 n}}{\left(2 n + 1\right)!}

Relations

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ζ ⁣(a,b)+ζ ⁣(b,a)=ζ ⁣(a)ζ ⁣(b)ζ ⁣(a+b)\zeta\!\left(a, b\right) + \zeta\!\left(b, a\right) = \zeta\!\left(a\right) \zeta\!\left(b\right) - \zeta\!\left(a + b\right)
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ζ ⁣(3,1,,3,1(3,1)  n times)=12n+1ζ ⁣(2,,22n times)\zeta\!\left(\underbrace{3, 1, \ldots, 3, 1}_{\left(3, 1\right) \; n \text{ times}}\right) = \frac{1}{2 n + 1} \zeta\!\left(\underbrace{2, \ldots, 2}_{2 n \text{ times}}\right)

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-04-08 16:14:44.404316 UTC