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

In this paper, we begin by reviewing the calculus induced framework of [10]. there, extended Polylogarithm functions over a subalgebra noncommutative rational power series, recognizable finite state (multiplicity) automata alphabet \(X=\{x_0,x_1\}\). The stability under shuffle products relies on nuclearity target space [32]. We also concentrated algebraic and analytic aspects extension allowing to index polylogarithms, at non positive multi-indices, series regularize divergent polyzetas, multi-indices As continuation works in [10] order understand bridge between “polylogarithmic calculus” world harmonic sums, propose local theory, adapted full indices Harmonic Sums based Taylor expansions, around zero, polylogarithms with \(x_1\) rightmost end. This theory is not only compatible Stuffle but Analytic Model. respect, it provides stable fully algorithmic model for calculus. Examples computer are provided.