2015/04/21 by David Broadhurst, Broadhurst, David
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #hep-th #math.NT
paper · pdf · doi:10.48550/arxiv.1504.05303
31 pages: version 2 includes a link to letter by Pierre Deligne
arxiv created 2015/04/24 · arxiv updated 2015/04/27
Multiple Landen values (MLVs) are defined as iterated integrals on the interval x∈[0,1] of the differential forms A=dlog(x), B=-dlog(1-x), F=-dlog(1-ρ2x) and G=-dlog(1-ρx), where ρ=(√(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 Tw, generated by 1/(1-x-x2- x3)=1+∑w>0Tw xw, and that a basis is provided by all the words in the \A,G\ sub-alphabet that neither end in A nor contain A3. 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 Apéry-type sums Aw=∑n>0(-1)n+1n-w/2n\choose n and 6 ladder-combinations of depth-1 polylogarithms \rm Liw(ρp)=∑n>0ρpnn-w with p∈\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.