Tests of conjectures on multiple Watson values

Abstract

I define multiple Watson values (MWVs) as iterated integrals, on the interval $x\in[0,1]$, of the 6 differential forms $A=d\log(x)$, $B=-d\log(1-x)$, $T=-d\log(1-z_1x)$, $U=-d\log(1-z_2x)$, $V=-d\log(1-z_3x)$ and $W=-d\log(1-z_4x)$, where $z_1=\gamma^2$, $z_2=\gamma/(1+\gamma)$, $z_3=\gamma^2/(1-\gamma)$ and $z_4=\gamma=2\sin(\pi/14)$ solves the cubic $(1-\gamma^2)(1-\gamma)=\gamma$. Following a suggestion by Pierre Deligne, I conjecture that the dimension of the space of ${\mathbb Z}$-linearly independent MWVs of weight $w$ is the number $D_w$ generated by $1/(1-2x-x^2-x^3)=1+\sum_{w>0}D_w x^w$. This agrees with 6639 integer relation searches, of dimensions up to $D_5+1=85$, performed at 2000-digit precision, for $w<6$.