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
| $2$ | $3$ | $4$ | |
|---|---|---|---|
| $1$ | $\omega(0,1)$ | $\omega(0,0,1,0)$ | |
| $3$ | $\omega(0,3)$ | $\omega(0,0,0,3)$ | |
| $5$ | $\omega(0,5)$ | $\omega(0,0,0,5),$ $\omega(0,0,2,3)$ | |
| $7$ | $\omega(0,7)$ | $\omega(0,0,0,7),$ $\omega(0,0,2,5),$ $\omega(0,0,4,3)$ | |
| $9$ | $\omega(0,9)$ | $\omega(0,0,0,9),$ $\omega(0,0,2,7),$ $\omega(0,0,4,5),$ $\omega(0,1,3,5)$ | |
| $11$ | $\omega(0,11)$ | $\omega(0,0,0,11),$ $\omega(0,0,2,9),$ $\omega(0,0,4,7),$ $\omega(0,1,3,7),$ $\omega(0,3,3,5)$ | |
| $13$ | $\omega(0,13)$ | $\omega(0,0,0,13),$ $\omega(0,0,2,11),$ $\omega(0,0,4,9),$ $\omega(0,0,6,7),$ $\omega(0,1,3,9),$ $\omega(0,1,5,7),$ $\omega(0,3,3,7),$ $\omega(0,3,5,5)$ |
Indecomposable eMZVs of even weight
| $2$ | $3$ | $4$ | $5$ | |
|---|---|---|---|---|
| $2$ | $\omega(0,0,2)$ | $\omega(0,0,0,0,2)$ | ||
| $4$ | $\omega(0,0,4)$ | $\omega(0,0,0,0,4),$ $\omega(0,0,0,1,3)$ | ||
| $6$ | $\omega(0,0,6)$ | $\omega(0,0,0,0,6),$ $\omega(0,0,0,1,5),$ $\omega(0,0,0,2,4),$ $\omega(0,0,2,2,2)$ | ||
| $8$ | $\omega(0,0,8),$ $\omega(0,3,5)$ | $\omega(0,0,0,0,8),$ $\omega(0,0,0,1,7),$ $\omega(0,0,0,2,6),$ $\omega(0,0,1,2,5),$ $\omega(0,0,2,2,4)$ | ||
| $10$ | $\omega(0,0,10),$ $\omega(0,3,7)$ | $\omega(0,0,0,0,10),$ $\omega(0,0,0,1,9),$ $\omega(0,0,0,2,8),$ $\omega(0,0,0,3,7)$ $\omega(0,0,1,2,7),$ $\omega(0,0,1,3,6),$ $\omega(0,0,1,4,5),$ $\omega(0,0,2,2,6),$ $\omega(0,0,2,2,4)$ | ||
| $12$ | $\omega(0,0,12),$ $\omega(0,3,9)$ | $\omega(0,0,0,0,12),$ $\omega(0,0,0,1,11),$ $\omega(0,0,0,2,10),$ $\omega(0,0,0,3,9),$ $\omega(0,0,0,4,8),$ $\omega(0,0,1,2,9),$ $\omega(0,0,1,3,8),$ $\omega(0,0,1,4,7),$ $\omega(0,0,1,5,6),$ $\omega(0,0,2,2,8),$ $\omega(0,0,2,4,6),$ $\omega(0,1,1,2,8)$ | ||
| $14$ | $\omega(0,0,14),$ $\omega(0,3,9),$ $\omega(0,5,11)$ | $\omega(0,0,0,0,14),$ $\omega(0,0,0,1,13),$ $\omega(0,0,0,2,12),$ $\omega(0,0,0,3,11),$ $\text{and many more}$ |
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)\,,