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% leave as is
\issue{ACA conference 2025}
\articlehead{}
\titlehead{Various products of representative series and some applications}
\authorhead{V. C. Bui}
\setcounter{page}{1}

\begin{document}

\title{Various products of representative series \\and some applications}

\author{Van Chien BUI \\
Department of Mathematics \\
University of Sciences, Hue University \\
Hue, Vietnam \\
\url{bvchien@hueuni.edu.vn}}

\date{\today}

\maketitle

\begin{abstract}
Special functions such as polyzetas, multiple harmonic sums and polylogarithms are defined over $\calH_r:=\{(s_1,\ldots,s_r)\in\N_{\ge1}^r,s_1>1\}$. Polyzetas values are given by the formula:
\begin{eqnarray}
	\zeta(s_1,\ldots,s_r)=\sum_{n_1>\ldots>n_r>0}{n_1^{-s_1}\ldots n_r^{-s_r}},
\end{eqnarray}
polylogarithms (denoted $(Li_{s_1, \dots, s_r})$ with $s_j \ge 1, r \ge 1$) and  multiple harmonic sums (denoted $(H_{s_1, \dots, s_r})$ with $s_j \ge 1, r \ge 1$). They are defined as follows (with $n \in \N_{\ge1}$):
\begin{equation}
	Li_{s_1, \dots, s_r}(z) = \sum_{n_1 > \dots > n_r > 0} n_1^{-s_1} \dots n_r^{-s_r} z^{n_1}
\end{equation}
and
\begin{equation}
	H_{s_1, \dots, s_r}(n) = \sum_{n \ge n_1 > \dots > n_r > 0} n_1^{-s_1} \dots n_r^{-s_r}.
	\label{eq:H_sum_corrected_en}
\end{equation}
They are compatible with algebraic structures of quasi-shuffle products, in some different cases of the parameter $q$:
\begin{eqnarray}\label{stuffle}
	u\stuffle 1_{Y^*}=1_{Y^*}\stuffle u=u,
	&y_iu\stuffle y_jv=y_i(u\stuffle y_jv)+y_j(y_iu\stuffle v)+qy_{i+j}(u\stuffle v),
\end{eqnarray}
where $\epsilon$ is the empty word,  $y_i,y_j, y_{i+j}$ are letters of the alphabet $Y=\{y_k\}_{k\in \N_{\ge1}}$, and $u,v$ are words in the monoid $Y^*$.
%

For a commutative ring $A$ containing the field of rational numbers
$\mathbb{Q}$, we examine the set of noncommutative formal series, denoted by $\ncs{A}{\calX} $. Within this set, representative series, that are closed under various products,  form a module. This is a central focus of our research. 

In this presentation, we will delve into how to factorize and decompose these noncommutative rational series and explore their relevance to theoretical computer science.

\end{abstract}

\begin{thebibliography}{99}
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\bibitem{BDM1}Bui, V. C., Duchamp, G. \& Hoang Ngoc Minh, V. Schützenberger's factorization on the (completed) Hopf algebra of q-stuffle product. {\em JP J. Algebra Number Theory Appl.}. \textbf{30}, 191-215 (2013)
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\bibitem{BDM3}Bui, V. C., Duchamp, G. \& Hoang Ngoc Minh, V. Computation tool for the q-deformed quasi-shuffle algebras and representations of structure of MZVs. {\em ACM Commun. Comput. Algebra}. \textbf{49}, 117-120 (2015)
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\bibitem{BDMTN}Chien, B., Duchamp, G., Minh, H., Tollu, C. \& Nghia, N. Combinatorics of $\phi$-deformed stuffle Hopf algebras. {\em CoRR}. \textbf{abs/1302.5391} (2013), http://arxiv.org/abs/1302.5391
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\end{thebibliography}

\end{document}
