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 .
翻译:在本文中,我们首先审查由 [10] 框架引发的微积分。 在该文件中, 我们将多元数函数扩展为非混合理性电源序列的子代数, 由字母X = {x 0, x 1 } 的有限状态( 多重) 自动成像来识别。 洗牌产品下的这种微积分的稳定性取决于目标空间的核度 [31] 。 我们还集中关注此扩展的代数和分析方面, 允许在非正数多指数、 理性序列以及允许在非正数多指数中调节不同的多元热量序列。 作为X X = {x 0, x 1 的有限状态( 多重) 的自动成象数 。 为了理解这个“ 多元性微积分” 的延伸与调量世界之间的桥梁, 我们提出了一种本地理论, 适应于基于泰勒扩展、 约零 、 6 多元数指数 和 数象数 等指数 的指数指数指数指数指数, 也提供了一种完全的计算机模型 。 。 这个模型与这个模型的模型的模型 也提供了 的完整的逻辑 。