Towards a Theory of Domains for Harmonic Functions and its Symbolic Counterpart

In this paper, we begin by reviewing the calculus induced by the framework of [ 10 ]. In there, we extended Polylogarithm functions over a subalgebra of noncommutative rational power series, recognizable by finite state (multiplicity) automata over the alphabet X = { x 0 , x 1 } . The stability of this calculus under shuffle products relies on the nuclearity of the target space [ 32 ]. We also concentrated on algebraic and analytic aspects of this extension allowing to index polylogarithms, at non positive multi-indices, by rational series and also allowing to regularize divergent polyzetas, at non positive multi-indices [ 10 ]. As a continuation of works in [ 10 ] and in order to understand the bridge between the extension of this “polylogarithmic calculus” and the world of harmonic sums, we propose a local theory, adapted to a full calculus on indices of Harmonic Sums based on the Taylor expansions, around zero, of polylogarithms with index x 1 on the rightmost end. This theory is not only compatible with Stuffle products but also with the Analytic Model. In this respect, it provides a stable and fully algorithmic model for Harmonic calculus. Examples by computer are also provided.