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In this paper, we begin by reviewing the calculus induced by the framework of[10]. In there, we extended Polylogarithm functions over a subalgebra ofnoncommutative rational power series, recognizable by finite state(multiplicity) automata over the alphabet X = {x 0 , x 1 }. The stability ofthis calculus under shuffle products relies on the nuclearity of the targetspace [31]. We also concentrated on algebraic and analytic aspects of thisextension allowing to index polylogarithms, at non positive multi-indices, byrational series and also allowing to regularize divergent polyzetas, at nonpositive multi-indices [10]. As a continuation of works in [10] and in order tounderstand the bridge between the extension of this "polylogarithmic calculus"and the world of harmonic sums, we propose a local theory, adapted to a fullcalculus on indices of Harmonic Sums based on the Taylor expansions, aroundzero, of polylogarithms with index x 1 on the rightmost end. This theory is notonly compatible with Stuffle products but also with the Analytic Model. In thisrespect, it provides a stable and fully algorithmic model for Harmoniccalculus. Examples by computer are also provided 6 .

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