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 $A=d\log (x)$, $B=-d\log (1-x)$, $T=-d\log (1-z_{1}x)$, $U=-d\log (1-z_{2}x)$, $V=-d\log (1-z_{3}x)$ and $W=-d\log (1-z_{4}x)$, 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 $w<6$.

7 pages

References in corpus (2)

• Multiple Deligne values: a data mine with empirically tamed denominators
• Multiple Landen values and the tribonacci numbers