arXiv subjects
Publications and source records attributed to Olivier Bouillot.
We define a two-parameter deformation of the quasi-shuffle by means of the formal group law associated with the exponential generating function of the homogeneous Eulerian polynomials, and construct bases of $QSym$ and $\WQSym$ whose product rule is given by this operation.
Multizeta values are numbers appearing in many different contexts. Unfortunately, their arithmetics remains mostly out of reach. In this article, we define a functional analogue of the algebra of multizetas values, namely the algebra of multitangent functions, which are 1- periodic functions defined by a process formally similar to multizeta values. We introduce here the fundamental notions of reduction into monotangent functions, projection onto multitangent functions and that of trifactorization, giving a way of writing a multitangent function in terms of Hurwitz multizeta functions. This explains why the multitangent algebra is a functional analogue of the algebra of multizeta values. We then discuss the most important algebraic and analytic properties of these functions and their consequences on multizeta values, as well as their regularization in the divergent case. Each property of multitangents has a pendant on the side of multizeta values. This allows us to propose new conjectures, which have been checked up to the weight 18.
Multizeta values are real numbers which span a complicated algebra: there exist two different interacting products. A functional analog of these numbers is defined so as to obtain a better understanding of them, the Hurwitz multizeta functions, which span an algebra for which a precise description is wanted. In this note, we prove that the algebra of Hurwitz multizeta functions is a polynomial algebra.
In this short Survey we revisit the subject of local, identity-tangent diffeomorphisms of $\doC$ and their analytic invariants, under two viewpoints: that of explicit expansions, which necessarily involve multitangents and multizetas; and that of effective computation. Along the way, we stress the difference between the \textup{collectors} (pre-invariant but of one piece) and the \textup{connectors} (invariant but mutually unrelated). We also attempt to streamline the nomenclature and notations.