# Multiple zeta values

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

## Definitions

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

## Sum representations

$\zeta\!\left({s}_{1}, {s}_{2}, \ldots, {s}_{k}\right) = \sum_{\textstyle{n \in {\mathbb{Z}}^{k} \atop {n}_{1} > {n}_{2} > \ldots > {n}_{k} > 0}} \prod_{i=1}^{k} \frac{1}{{n}_{i}^{{s}_{i}}}$

## Specific values

$\zeta\!\left(2, 1\right) = \sum_{n=1}^{\infty} \frac{H_{n}}{{\left(n + 1\right)}^{2}} = \zeta\!\left(3\right)$
$\zeta\!\left(2, 2\right) = \frac{3}{4} \zeta\!\left(4\right)$
$\zeta\!\left(3, 2\right) = 3 \zeta\!\left(2\right) \zeta\!\left(3\right) - \frac{11}{2} \zeta\!\left(5\right)$
$\zeta\!\left(4, 2\right) = {\left(\zeta\!\left(3\right)\right)}^{2} - \frac{4}{3} \zeta\!\left(6\right)$
$\zeta\!\left(2, 3\right) = \frac{9}{2} \zeta\!\left(5\right) - 2 \zeta\!\left(2\right) \zeta\!\left(3\right)$
$\zeta\!\left(3, 3\right) = \frac{1}{2} \left({\left(\zeta\!\left(3\right)\right)}^{2} - \zeta\!\left(6\right)\right)$

## Families of closed forms

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

## Relations

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

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

2021-03-15 19:12:00.328586 UTC