Multiple Landen values and the tribonacci numbers

Abstract

Multiple Landen values (MLVs) are defined as iterated integrals on the interval $x\in[0,1]$ of the differential forms $A=d\log(x)$, $B=-d\log(1-x)$, $F=-d\log(1-\rho^2x)$ and $G=-d\log(1-\rho x)$, where $\rho=(\sqrt{5}-1)/2$ is the golden section. I conjecture that the dimension of the space of ${\mathbb Z}$-linearly independent MLVs of weight $w$ is a tribonacci number $T_w$, generated by $1/(1-x-x^2- x^3)=1+\sum_{w>0}T_w x^w$, and that a basis is provided by all the words in the $\{A,G\}$ sub-alphabet that neither end in $A$ nor contain $A^3$. For $w<9$, I construct a much more efficient basis, for a MLV datamine, where no prime greater than 11 occurs in the denominators of 3,357,257 coefficients of rational reduction of 49,151 MLVs. Numerical data for 40 primitives then enable fast evaluation of all of these MLVs to 20,000 digits. The datamine provides reductions of Ap\'ery-type sums $A_w=\sum_{n>0}(-1)^{n+1}n^{-w}/{2n\choose n}$ and 6 ladder-combinations of depth-1 polylogarithms ${\rm Li}_w(\rho^p)=\sum_{n>0}\rho^{pn}n^{-w}$ with $p\in\{1,2,3,4,6,8,10,12,20,24\}$ and coefficients given by Landen, Coxeter and Lewin at $w=2$. I prove that the former evaluate to MLVs and conjecture that the latter do. Comparison is made between the properties of MLVs and multiple polylogarithms at roots of unity, encountered in the quantum field theory of the standard model of particle physics.