elliptic multiple zeta values

Relations

In the following tables we display our (non-canonical) choice of indecomposable eMZVs for a given weight and length. Further below, explicit rules allowing to express eMZVs in terms of those are provided. The appearance of usual MZVs is discussed in [arXiv:1507.02254].

Indecomposable eMZVs of odd weight

	$2$	$4$
$1$	$\omega(0,1)$	$\omega(0,0,1,0)$
$3$	$\omega(0,3)$	$\omega(0,0,0,3)$
$5$	$\omega(0,5)$	$\omega(0,0,0,5),$ $\omega(0,0,2,3)$
$7$	$\omega(0,7)$	$\omega(0,0,0,7),$ $\omega(0,0,2,5),$ $\omega(0,0,4,3)$
$9$	$\omega(0,9)$	$\omega(0,0,0,9),$ $\omega(0,0,2,7),$ $\omega(0,0,4,5),$ $\omega(0,1,3,5)$
$11$	$\omega(0,11)$	$\omega(0,0,0,11),$ $\omega(0,0,2,9),$ $\omega(0,0,4,7),$ $\omega(0,1,3,7),$ $\omega(0,3,3,5)$
$13$	$\omega(0,13)$	$\omega(0,0,0,13),$ $\omega(0,0,2,11),$ $\omega(0,0,4,9),$ $\omega(0,0,6,7),$ $\omega(0,1,3,9),$ $\omega(0,1,5,7),$ $\omega(0,3,3,7),$ $\omega(0,3,5,5)$

Indecomposable eMZVs of even weight

	$3$	$5$
$2$	$\omega(0,0,2)$	$\omega(0,0,0,0,2)$
$4$	$\omega(0,0,4)$	$\omega(0,0,0,0,4),$ $\omega(0,0,0,1,3)$
$6$	$\omega(0,0,6)$	$\omega(0,0,0,0,6),$ $\omega(0,0,0,1,5),$ $\omega(0,0,0,2,4),$ $\omega(0,0,2,2,2)$
$8$	$\omega(0,0,8),$ $\omega(0,3,5)$	$\omega(0,0,0,0,8),$ $\omega(0,0,0,1,7),$ $\omega(0,0,0,2,6),$ $\omega(0,0,1,2,5),$ $\omega(0,0,2,2,4)$
$10$	$\omega(0,0,10),$ $\omega(0,3,7)$	$\omega(0,0,0,0,10),$ $\omega(0,0,0,1,9),$ $\omega(0,0,0,2,8),$ $\omega(0,0,0,3,7)$ $\omega(0,0,1,2,7),$ $\omega(0,0,1,3,6),$ $\omega(0,0,1,4,5),$ $\omega(0,0,2,2,6),$ $\omega(0,0,2,2,4)$
$12$	$\omega(0,0,12),$ $\omega(0,3,9)$	$\omega(0,0,0,0,12),$ $\omega(0,0,0,1,11),$ $\omega(0,0,0,2,10),$ $\omega(0,0,0,3,9),$ $\omega(0,0,0,4,8),$ $\omega(0,0,1,2,9),$ $\omega(0,0,1,3,8),$ $\omega(0,0,1,4,7),$ $\omega(0,0,1,5,6),$ $\omega(0,0,2,2,8),$ $\omega(0,0,2,4,6),$ $\omega(0,1,1,2,8)$
$14$	$\omega(0,0,14),$ $\omega(0,3,9),$ $\omega(0,5,11)$	$\omega(0,0,0,0,14),$ $\omega(0,0,0,1,13),$ $\omega(0,0,0,2,12),$ $\omega(0,0,0,3,11),$ $\text{and many more}$

Length 2

A closed formula is available for length two:

$\,\omega(n_1,n_2) \, \Big|_{n_1+n_2 \ \text{odd}} = (-1)^{n_1} \,\omega(0,n_1+n_2) + 2 \,\delta_{n_1,1} \,\zeta_{n_2} \,\omega(0,1) - 2 \,\delta_{n_2,1} \,\zeta_{n_1} \,\omega(0,1)$

$+2 \sum_{p=1}^{\lceil \frac{1}{2}(n_2-3) \rceil} {n_1+n_2 - 2p-2\choose n_1-1} \,\zeta_{n_1+n_2-2p-1} \,\omega(0,2p+1)$

$-2\sum_{p=1}^{\lceil \frac{1}{2}(n_1-3) \rceil} {n_1+n_2 - 2p-2\choose n_2-1} \,\zeta_{n_1+n_2-2p-1} \,\omega(0,2p+1)\,,$

Relations

Indecomposable eMZVs of odd weight

Indecomposable eMZVs of even weight

Length 2

Length 3

Length 4

Length 5

Length 6