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 |
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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. |
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ISSN: | 1661-8270 1661-8289 |
DOI: | 10.1007/s11786-022-00552-5 |