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\titlehead{Multiplicative structure of some multivariate functions}
\authorhead{Jean-Yves Enjalbert}
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\begin{document}

\title{Multiplicative structure of some multivariate functions}

\author{Jean-Yves Enjalbert}

\date{}

\maketitle

\section{Introduction}\label{section-introduction}

It is now well known that the famous Riemann zeta function admits a multivariate extension given by
$$\zeta(s_1;\ldots;s_n)=\sum_{n_1>\ldots>n_r}\dfrac{1}{n_1^{s_1}\ldots n_r^{s_r}}$$
with, initially, $s_1,\ldots,s_r$ being natural numbers.
This family of functions is studied for both its arithmetic \cite{Waldschmidt1} and computational interest, particularly in quantum physics. To obtain this, an efficient coding approach has been proposed using two-letter or $n$-letter alphabets.
We will generalize this process to other families of functions: colored polyzetas, {\em reals} Hurwitz polyzetas \cite{ThNb}, multifunctional Polylerch, etc.
For each of these families, we explain how their multiplicative structures relate via their coding to the $\varphi$-shuffle, defined recursively by:
\begin{equation}
\forall(a,b)\in X^2, \forall (u,v)\in \left(X^*\right)^2,\quad
au\pshuffle{\varphi} vb=a(u\pshuffle{\varphi} bv)+b(au\pshuffle{\varphi} v)+\varphi(a,b)(u\pshuffle{\varphi} v)\ ,
\end{equation}
thus providing an implementable computational path.
We will then begin a systematic study of $\varphi$-shuffles.

\section{II- different type of $\varphi$-shuffle}

In this study, we will be led to distinguish five types of $\varphi$-shuffle :
\begin{enumerate}
	\item Type I : factor $\varphi$ comes from a product (possibly with zero) between letters (i.e. $X\cup \{0\}$ is a semigroup).
	\item Type II : factor $\varphi$ comes from the deformation of a semigroup product by a bicharacter.
	\item Type III : factor $\varphi$ comes from the deformation of a semigroup product by a colour factor.
	\item Type IV : factor $\varphi$ is the commutative law of an associative algebra (CAA) on $A.X$
	\item Type V : factor $\varphi$ is the law of an associative algebra (AA) on $A.X$
\end{enumerate}
%
These classes are ordered by the following (strict) inclusion diagram:
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     &    &  \mbox{IV} &\\
};
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\caption{Hasse diagram of the inclusions between classes.}\label{classes_ord}
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\newpage
We have collected examples from the literature, with the corresponding formulas, and give a classication  table.%Associated formulas and illustrations in literature are given in the following table. 


\section{III - Radford's theorem for AC-shuffle}

Once the multiplicative laws are transferred via coding, it becomes easier to work on the alphabet $X$. This alphabet has a total order relation $<$, we then have the family $\Lyn(X)$ of Lyndon words on $X$ \cite{Reutenauer}. We will note the decompositions based on $\Lyn(X)$ as follows:

\begin{definition}\label{multidegree_notation}
Let $\star$  be an associative law, with unit, over $\ncp{A}{X}$. For any $\alpha\in \N^{(Lyn(X))}$ and $\{l_1,\cdots ,l_r\}\supset supp(\alpha)$ in strict decreasing order (i.e. $l_1>\cdots >l_r$), we set 
\begin{equation}\label{multprod}
\X^{\star\alpha}= l_1^{\star\alpha_1}\star\cdots \star l_r^{\star\alpha_r},
\end{equation}  
where $\alpha_i=\alpha(l_i)$ for all $i$ and, for short, $\X=Lyn(X)$.
\end{definition}
We will establish the following theorem :
\begin{theorem}\label{th:Lyn-linbasis} 
Let $A$ be a commutative ring (with unit) such that\footnote{This condition amounts to ask that $\N^+.1_A\subset A^\times$} $\Q\subset A$ and $\pshuffle{\varphi} : \ncp{A}{X}\otimes \ncp{A}{X}\rightarrow \ncp{A}{X}$ is associative.\\
If $X$ is totally ordered by $<$, then $\left(\X^{\pshuffle{\varphi}\alpha}\right)_{\alpha\in\N^{(\Lyn(X))}}$ is a linear basis of $\ncp{A}{X}$.
\end{theorem}
So, $\Lyn(X)$ is a transcendence basis of $\mathcal{A}=(\ncp{A}{X},\pshuffle{\varphi},1_{X^*})$.

\section{IV -Bialgebra structure ... in the way of Hofp algebra}

It turns out to be very efficient to dualize, if possible, the algebra $\mathcal{A}=(\ncp{A}{X},\pshuffle{\varphi},1_{X^*})$ in the following way :
\begin{definition}
A law $\star$ defined over $\ncp{A}{X}$ is a dual law (or dualizable) if there exists   
a linear mapping 
$\Delta_{\star}:\ncp{A}{X}\rightarrow\ncp{A}{X}\otimes\ncp{A}{X}$ such 
\begin{eqnarray}
\forall(u,v,w)\in X^*\times X^*\times X^*, \quad
\scal{u\star v}{w}=\scal{u\otimes v}{\Delta_{\star}(w)}^{\otimes 2}\ .
\end{eqnarray}
In this case, $\Delta_*$ will be called the comultiplication dual to $\star$.
\end{definition}
Here again, exactly the same conditions allow dualization :
{
\begin{theorem}\label{bialgebra} Let $A$ be a commutative ring (with unit). We suppose that  the product $\shuffle_{\varphi} : \ncp{A}{X}\otimes \ncp{A}{X}\rightarrow \ncp{A}{X}$ is an associative and commutative law on $\ncp{A}{X}$, then
the algebra $(\ncp{A}{X},\pshuffle{\varphi},1_{X^*})$ can be endowed with 
the comultiplication $\Delta_{\conc}$ dual to the concatenation
\begin{equation}
	\Delta_{\conc}(w)=\sum_{uv=w} u\otimes v
\end{equation}
and the ``constant term'' character $\epsilon(P)=\scal{P}{1_{X^*}}$.% as defined previously.\\ 
\begin{enumerate}
  \item[(i)] With this setting
\begin{equation}\label{b1}
	\mathcal{B}_{\varphi}=(\ncp{A}{X},\pshuffle{\varphi}, 1_{X^*}, \Delta_{\conc},\epsilon) 
\end{equation}
is a bialgebra \footnote{Commutative and, when $|X|\geq 2$, noncocommutative.}.
\item[(ii)] The bialgebra \mref{b1} is, in fact, a Hopf Algebra.
  \end{enumerate}
\end{theorem}
So we need to find the condition over $\pshuffle{\varphi}$ to be commutative and associative. We will given :
\begin{theorem}\label{th:prop_law}
\begin{itemize}
\item[(i)] The law $\pshuffle{\varphi}$ is commutative if and only if the extension $\varphi : AX\otimes AX\rightarrow AX$ is commutative. 
\item[(ii)] The law $\pshuffle{\varphi}$ is associative if and only if the extension $\varphi : AX\otimes AX\rightarrow AX$ is associative.
\end{itemize}
\end{theorem}
\begin{proposition}
We call $\gamma_{x,y}^z:=\scal{\varphi(x,y)}{z}$ the structure constants of $\varphi$ (w.r.t. the basis $X$).\\ 
The product $\pshuffle{\varphi}$ is a dual law if and only if $(\gamma_{x,y}^z)_{x,y,z\in X}$ is dualizable in the following sense
\begin{eqnarray}
	(\forall z\in X)(\#\{(x,y)\in X^2|\gamma_{x,y}^z\not=0\}<+\infty)\ .
\end{eqnarray}
\end{proposition}

\subsection{The Hopf-Hurwitz algebra}

In the end we explicitly give the product $\luffle$ of {\em reals} Hurwitz polyzetas, and check  its commutativity and associativity, making $(\ncp{A}{N},\luffle,1_N)$ into a $A$-CAAU : the  Radford's theorem can be generalised here.




\begin{thebibliography}{9}

\bibitem{Waldschmidt1}{M.~Waldschmidt}.--
        {\em Valeurs z\^eta multiples~: une introduction},
        Journal de Th\'eorie des Nombres de Bordeaux, \textbf{12}.2, 2000, pages 581-595.
      
\bibitem{Hoffman}{ M.E.~Hoffman}.--
{\em   Quasi-shuffle products}, 
Journal of Algebraic Combinatorics, 11 (2000), 49-68.
volume \textbf{11},
2000, 
pages 49-68.

\bibitem{Reutenauer}{C.~Reutenauer C}.-- 
{\em Free Lie Algebras},
Clarendon Press (London Mathematical Society Monographs, vol. 7, Oxford, 1993.
 %
 
 \bibitem{Weinzierl}{S.~Weinzierl}.--
{\em Hopf algebras and Dyson-Schwinger equations},
Lectures given at the workshop ''Dyson-Schwinger Equations in Modern Physics and Mathematics'',Trento,September 2014.


\bibitem{ThNb}{J.Y.-Enjalbert  and  V.~Hoang Ngoc Minh V}.-- 
{\em Analytic and combinatoric aspects of Hurwitz polyz\^etas}, 
Journal Th\'eorie des Nombres de Bordeaux, 
volume \textbf{23}(2), 2011,
pages 353-386.

\end{thebibliography}

\end{document}
