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

Abstract : 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 [31]. 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 6 .
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https://hal.archives-ouvertes.fr/hal-03401583
Contributor : Gérard Duchamp Connect in order to contact the contributor
Submitted on : Monday, October 25, 2021 - 1:54:25 PM
Last modification on : Wednesday, October 27, 2021 - 4:05:06 AM

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  • HAL Id : hal-03401583, version 1
  • ARXIV : 2110.13743

Citation

van Chiên Bui, Gérard Duchamp, Quoc Hoàn Ngo, Vincel Hoang Ngoc Minh, Vu Nguyen Dinh. Towards a Theory of Domains for Harmonic Functions and its Symbolic Counterpart.. 2021. ⟨hal-03401583⟩

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