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\titlehead{Families of eulerian functions involved in regularization of divergent polyzetas}
\authorhead{Ngo Quoc Hoan}
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\title{Families of eulerian functions involved in regularization of divergent polyzetas}

\author{Ngo Quoc Hoan \\
Faculty of  Mathematics and Informatics, \\ Hanoi University of Science and Technology,\\
1 Dai Co Viet, Ha Noi, Viet Nam\\ 
\url{quochoan_ngo@yahoo.com.vn}}

\date{}

\maketitle

\begin{abstract}
%\section{Introduction}
For any $r\in\mathbb{N}_{\ge1}$ and $(s_1,\ldots,s_r)\in{\mathbb C}^r$,
let us consider the following {\it several variable zeta function} (polyzetas) \cite{IMACS}
$
\zeta(s_1,\ldots,s_r):=\sum\limits_{n_1>\ldots>n_r>0}n_1^{-s_1}\ldots n_r^{-s_r}
$
which converges for $(s_1,\ldots,s_r)$ in the open sub-domain of $\mathbb{C}^r$ \cite{Goncharov, Zhao},
$
\mathcal{H}_r:=\{(s_1,\ldots,s_r)\in \mathbb{C}^r\,\vert\,
\forall m=1,\ldots,r,\,  \sum\limits_{ i =1}^{m} \mathrm{Re}(s_i) > m\}.
$
%The function $\Gamma$ is meromorphic, with no zeroes and $-\mathbb{N}^*$ as set of simple poles.
%Hence $\Gamma^{-1}$ is entire and admits $-\mathbb{N}^*$ as set of simple zeroes. Moreover,
%it satisfies\footnote{{\it i.e.} its coefficients are real, we will see later the
%combinatorial content of them.} $\Gamma(\overline{z})=\overline{\Gamma(z)}$.
From Weierstrass factorization  and Newton-Girard identity \cite{IMACS, GHM},
one has successively
\begin{eqnarray}\label{Weierstrass}
\frac1{\Gamma(z+1)}
=e^{\gamma z}\prod_{n\ge1}\Bigl(1+\frac{z}{n}\Bigr)e^{-\frac{z}{n}}
=\exp\Bigl(\gamma z-\sum_{k\ge2}\zeta(k)\frac{(-z)^k}{k}\Bigr)
\end{eqnarray}
where $\Gamma (z)$ defines  the Gamma function.
One can deduce the following expression for ${\zeta(2k)}$:
\begin{eqnarray}\label{paire}
\frac{{\zeta(2k)}}{\pi^{2k}}= k\sum_{l=1}^k\frac{(-1)^{k+l}}{l}
\sum_{n_1,\ldots,n_l\ge1\atop n_1+\ldots+n_l=k}\prod_{i=1}^l\frac{1}{{\Gamma(2n_i+2)}}\in\mathbb{Q}.
\end{eqnarray}
The formula \eqref{paire} is a different version of a result of L. Euler using Bernoulli numbers
\begin{eqnarray*}
\dfrac{\zeta(2k)}{\pi^{2k}} = \dfrac{(-1)^{k+1}2^{2k-1} B_{2k} }{(2k)!}, \ k \in \mathbb{N}.
\end{eqnarray*}
In this talk, based on the combinatorics of noncommutative
generating series, we discuss a way to extend the formula \eqref{Weierstrass}  and then we present a recurrence relation of $\zeta (2^k, \ldots , 2^k),  \ k \in \mathbb{N}^*$. This is based on join works with Prof. Hoang Ngoc Minh and Prof. Gérard Duchamp \cite{GHM}.
%In the paper \cite{CholeskyWork:2006} $\ldots$
%short description of the paper
\end{abstract}


\bibliographystyle{amsplain}
\begin{thebibliography}{99}
\bibitem{Dieudonne}{Jean Dieudonn\'e}, \textit{Infinitesimal calculus}, Houghton Mifflin, 1971.
\bibitem{Goncharov}{A.B. Goncharov}, \textit{Multiple polylogarithms and mixed Tate motives}, 2001.
%\bibitem{Hoan}{Ngo Quoc Hoan}, \textit{The value of polyzetas }, submitted,  2023.
\bibitem{IMACS}{V. Hoang Ngoc Minh}, \textit{Summations of Polylogarithms via Evaluation Transform},
in Math. \& Comp. in Simul., 1336, p. 707-728, 1996.
\bibitem{GHM}{Bui Van Chien, Hoang Ngoc Minh, Ngo Quoc Hoan, Nguyen Dinh Vu}, \textit{Families of eulerian functions involved in regularization of divergent polyzetas},  Pub. Math. de Besancon, p. 5-28, 2023.
\bibitem{lascoux}{A. Lascoux}, \textit{Fonctions sym\'etriques}, SLC, B08e, 1983.
\bibitem{Zhao}{J. Zhao}, \textit{Analytic continuation of multiple zeta functions}, Proc. A. M. S., 128 (5), p. 1275 - 1283, 1999.
\end{thebibliography}

%\begin{thebibliography}{99}

%\bibitem{CholeskyWork:2006}
%Claude Brezinski.
%\newblock The life and work of {A}ndr\'e {C}holesky.
%\newblock {\em Numer. Algorithms}, 43:279--288, 2006.

%\end{thebibliography}

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