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 t...

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Veröffentlicht in:Mathematics in computer science 2023-03, Vol.17 (1), Article 4
Hauptverfasser: Chien, Bui Van, Duchamp, Gérard H. E., Hoan, Ngo Quoc, Vincel, Hoang Ngoc Minh, Vu, Nguyen Dinh
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Sprache:eng
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Zusammenfassung: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.
ISSN:1661-8270
1661-8289
DOI:10.1007/s11786-022-00552-5