2011/05/30 by Jakob Ablinger, Johannes Blümlein, Carsten Schneider · 1 citation
Physics and Astronomy · Computer Science · Mathematics · #math-ph #cs.SC #hep-ph #hep-th #math.AG #math.MP
paper · pdf · doi:10.1063/1.3629472
55 pages, 1 figure, 1 style file
arxiv created 2011/05/30 · arxiv updated 2015/05/28
The computation of Feynman integrals in massive higher order perturbative calculations in renormalizable Quantum Field Theories requires extensions of multiply nested harmonic sums, which can be generated as real representations by Mellin transforms of Poincaré--iterated integrals including denominators of higher cyclotomic polynomials. We derive the cyclotomic harmonic polylogarithms and harmonic sums and study their algebraic and structural relations. The analytic continuation of cyclotomic harmonic sums to complex values of N is performed using analytic representations. We also consider special values of the cyclotomic harmonic polylogarithms at argument x=1, resp., for the cyclotomic harmonic sums at N → ∞, which are related to colored multiple zeta values, deriving various of their relations, based on the stuffle and shuffle algebras and three multiple argument relations. We also consider infinite generalized nested harmonic sums at roots of unity which are related to the infinite cyclotomic harmonic sums. Basis representations are derived for weight \sf w = 1,2 sums up to cyclotomy \sf l = 20.