13 research outputs found
Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials
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 Poincar\'e--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 is performed using analytic representations. We also consider
special values of the cyclotomic harmonic polylogarithms at argument ,
resp., for the cyclotomic harmonic sums at , 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}.Comment: 55 pages, 1 figure, 1 style fil