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\date{\textcolor{red}{Workshop QUANTUM and COMBINATORICS,\\
Zakopane (Tatra Mountains, Poland), 12-14 November 2009.}\\
(joint work with G.H.E.~Duchamp, Universit\'e Paris-Nord-LIPN)}

\title{Polyz\^etas}

\author{Hoang Ngoc Minh,\\
Universit\'e Lille 2-LIPN,\\
1, Place D\'eliot, 59024 Lille C\'edex,
France.}

\begin{document}

\maketitle

\begin{slide}{Summary}
\begin{enumerate}
\item Introduction.
\item Infinitesimal Braid relations and structures of polylogarithms,
\begin{enumerate}
\item From infinitesimal braid relations to $KZ$ system to structure of polylogarithms,
\item Polynomial relations among polyz\^etas by computer.
\end{enumerate}
\item Noncommutative generating series technology,
\begin{enumerate}
\item Double regularization to $\textcolor{red}{0}$,
\item Results {\it\`a la} Abel and generalized Euler constants,
\item Regularization to $\textcolor{red}{\gamma}$,
\item Action of the differential Galois group of polylogarithms on their asymptotic expansions.
\item Structure of polyz\^etas.
\item Towards the transcendence of $\textcolor{red}{\gamma}$.
\end{enumerate}
\item Coloured polyz\^etas.
\end{enumerate}
\end{slide}


\begin{slide}{}
\vfill
\centerline{{\Large INFINITESIMAL BRAID RELATIONS}}

\centerline{{\Large $\&$}}

\centerline{{\Large STRUCTURE OF POLYLOGARITHMS}}
\vfill
\end{slide}

\begin{slide}{Infinitesimal~braid~relation~\&~Knizhnik-Zamolodchikov~syst.}
In 1986, Drinfel'd introduced the differential system
associated to the Lie algebra of pure braid groups $\calT_n$~:
\begin{eqnarray*}
\Omega_n(z_1,\ldots,z_n)
=\sum_{1\le i<j\le n}\frac{t_{i,j}}{2{\rm i}\pi}\;\omega_{i,j}(z_1,\ldots,z_n)
=\sum_{1\le i<j\le n}\frac{t_{i,j}}{2{\rm i}\pi}\frac{d(z_i-z_j)}{z_i-z_j}.
\end{eqnarray*}
The flatness condition, $d\Omega_n-\Omega_n\wedge\Omega_n=0$, is equivalent
to the infinitesimal braid relation generated by $\{t_{i,j}\}_{1\le i,j\le n}$~:
\begin{eqnarray*}
t_{i,j}=0\phantom{_{j,i}}&\mbox{for}&i=j,\\
t_{i,j}=t_{j,i}&\mbox{for}&i\neq j,\\[0pt]
[t_{i,j},t_{i,k}+t_{j,k}]=0\phantom{_{j,i}}&\mbox{for distinct}&i,j,k,\\[0pt]
[t_{i,j},t_{k,l}]=0\phantom{_{j,i}}&\mbox{for distinct}&i,j,k,l.
\end{eqnarray*}
Hence, the differential system $KZ_n$
\begin{eqnarray*}\label{KZ}
d F(z_1,\ldots,z_n)=\Omega_n(z_1,\ldots,z_n)F(z_1,\ldots,z_n)
\end{eqnarray*}
is completely integrable over $\{(z_1,\ldots,z_n)\in\C^n|z_i\neq z_j\mbox{ for }i\neq j\}$.
\end{slide}

\begin{slide}{Examples}
$\bullet\quad\calT_2=\{t_{1,2}\}$.
\begin{eqnarray*}
\Omega_2(z)=\frac{t_{1,2}}{2{\rm i}\pi}\frac{d(z_1-z_2)}{z_1-z_2},&&
F(z_1,z_2)=(z_1-z_2)^{t_{1,2}/2{\rm i}\pi}.
\end{eqnarray*}

\vfill

$\bullet\quad\calT_3=\{t_{1,2},t_{1,3},t_{2,3}\},\ [t_{1,3},t_{1,2}+t_{2,3}]=[t_{2,3},t_{1,2}+t_{1,3}]=0$.
\begin{eqnarray*}
\Omega_3(z)\!\!\!&=&\!\!\!\frac{1}{2{\rm i}\pi}
\biggl[t_{1,2}\frac{d(z_1-z_2)}{z_1-z_2}+t_{1,3}\frac{d(z_1-z_3)}{z_1-z_3}
+t_{2,3}\frac{d(z_2-z_3)}{z_2-z_3}\biggr].\\
F(z_1,z_2,z_3)\!\!\!&=&\!\!\!G\biggl(\frac{z_1-z_2}{z_1-z_3}\biggr)
(z_1-z_3)^{(t_{1,2}+t_{1,3}+t_{2,3})/2{\rm i}\pi},
\end{eqnarray*}
where $G$ satisfies the following differential equation
\begin{eqnarray*}\label{drinfeld}
(DE)&&dG=(x_0\;\omega_0(z)+x_1\;\omega_1)G,\\
\mbox{with}&&
x_0:=\phantom{-}\frac{t_{1,2}}{2{\rm i}\pi},\qquad\omega_0(z):=\Frac{dz}{z}\\
\mbox{and}&&x_1:=-\frac{t_{2,3}}{2{\rm i}\pi},\qquad\omega_1(z):=\Frac{dz}{1-z}.
\end{eqnarray*}
\end{slide}

\begin{slide}{Iterated integral along a path and dilogarithms}

Let $\omega_0(z)={dz}/{z}$ et $\omega_1(z)={dz}/(1-z)$.
The \textcolor{red}{iterated integral} over $\omega_0,\omega_1$
associated to $w=x_{i_1}\cdots x_{i_{k}}\in X^*$ is defined by
\begin{eqnarray*}
\alpha_{z_0}^z(\eps)=1&\mbox{and}&
\alpha_{z_0}^z(x_{i_1}\ldots x_{i_k})=\int_{z_0}^{z}\ldots\int_{z_0}^{z_{k-1}}\omega_{i_1}(z_1)\ldots\omega_{i_k}(z_k).
\end{eqnarray*}
\begin{example}
\vskip-15.3mm
\begin{eqnarray*}
\alpha_0^z(x_0x_1)
&=&\int_0^z\int_0^s\omega_0(s)\omega_1(t)\\
&=&\int_0^z\int_0^s\Frac{ds}s\Frac{dt}{1-t}\\
&=&\int_0^z\Frac{ds}s\int_0^s{dt}\sum_{k\ge0}t^k\\
&=&\sum_{k\ge1}\int_0^z{ds}\frac{s^{k-1}}k\\
&=&\sum_{k\ge1}\frac{z^k}{k^2}\\
&=&\Li_2(z).
\end{eqnarray*}
\end{example}

\end{slide}

\begin{slide}{Polylogarithms, harmonic sums and polyz\^etas}

Classic cases ($N\in\mathbb{N}_+,r>0$ and $|z|<1$)~:
\begin{eqnarray*}
\Li_r(z)=\alpha_0^z(x_0^{n-1}x_1)=\sum_{n\ge1}\frac{z^n}{n^r}&\mbox{and}&\H_r(N)=\sum_{n=1}^{N}\frac1{n^r}.
\end{eqnarray*}

Generalization on multi-indices $\mathbf{s}=(s_1,\ldots,s_r)$~:
\begin{eqnarray*}
\alpha_0^z(x_0^{s_1-1}x_1\ldots x_0^{s_r-1}x_1)
=\Li_{\mathbf{s}}(z)=\Sum_{n_1>\ldots>n_r>0}\frac{z^{n_1}}{n_1^{s_1}\ldots n_r^{s_r}},&&\\
\P_{\mathbf{s}}(z)=\frac{\Li_{\mathbf{s}}(z)}{1-z}=\Sum_{N\ge0}\H_{\mathbf{s}}(N)\;z^N,
\mbox{ where }\H_{\mathbf{s}}(N)=\sum_{n_1>\ldots>n_r=1}^N\frac1{n_1^{s_1}\ldots n_r^{s_r}}.&&
\end{eqnarray*}

If $\textcolor{red}{s_1>1}$, by an Abel's theorem, then one has
\centerline{$\Lim_{z \rightarrow 1}\Li_{\mathbf{s}}(z)
=\Lim_{N \rightarrow \infty}\H_{\mathbf{s}}(N)
=\zeta(\mathbf{s})
=\Sum_{n_1>\ldots>n_r>0}\!\!\frac1{n_1^{s_1}\ldots n_r^{s_r}}$}\\
else $\textcolor{red}{?}$
\end{slide}

\begin{slide}{Euler-Maclaurin sommation formula}
$\Sum_{n=1}^{N}\Frac1{n}=\log N+\textcolor{red}{\gamma}
-\Sum_{j=1}^{k-1}\Frac{B_j}{j}\Frac1{N^j}+O\biggl(\Frac1{N^k}\biggr)$,\\
$\Sum_{n=1}^{N}\frac1{n^r}=\textcolor{red}{\zeta(r)}-\Frac{N^{1-r}}{(r-1)}
-\Sum_{j=r}^{k-1}\frac{B_{j-r+1}}{j-r+1}\binom{k-1}{j-1}\Frac1{N^j}+O\biggl(\Frac1{N^k}\biggr)$.

\vfill

Are $\textcolor{red}{\gamma}$ and $\zeta(r)$ algebraically independent~?

\vfill

Let $\calZ$ be the $\Q$-algebra generated by convergent polyz\^etas.\\
Let $\calZ'$ be the $\Q[\textcolor{red}{\gamma}]$-algebra generated by convergent polyz\^etas.

\vfill

How to determine the asymptotic expansion and the constants
associated to the \textcolor{red}{divergents} harmonic sums of the form
\begin{eqnarray*}
\H_{\{1\}^r}(N)
&=&\Sum_{N\ge n_1>\ldots>n_r>0}^{}\frac1{n_1\ldots n_r}?\\
\H_{\{1\}^k,\underbrace{s_{k+1}}_{>1},\ldots,s_r}(N)
&=&\Sum_{N\ge n_1>\ldots>n_r>0}^{N}
\frac1{n_1\ldots n_kn_{k+1}^{s_{k+1}}\ldots n_r^{s_r}}?
\end{eqnarray*}
Do there exists a generalization of $\textcolor{red}{\gamma}$~?
\end{slide}

\begin{slide}{Encoding the multi-indices by words}
\small
$Y=\{y_k|k\in\N_+\}$ ($y_1<y_2<\ldots$) and $X=\{x_0,x_1\}$ ($x_0<x_1$).\\
$Y^*$ (resp. $X^*$)~:~monoide generated by $Y$ (resp. $X$).
$$\mathbf{s}=(s_1,\ldots,s_r)\leftrightarrow
w=y_{s_1}\ldots y_{s_r}\leftrightarrow
w=x_0^{s_1-1}x_1\ldots x_0^{s_r-1}x_1.$$
$u$ and $v$ are \textcolor{red}{convergent} if $s_1>1$.
A word \textcolor{red}{divergent} is of the form
\begin{eqnarray*}
(\{1\}^k,s_{k+1},\ldots,s_r)
\leftrightarrow y_1^ky_{s_{k+1}}\ldots y_{s_r}
\leftrightarrow x_1^kx_0^{s_{k+1}-1}x_1\ldots x_0^{s_r-1}x_1,&\mbox{for}&k\ge1.
\end{eqnarray*}
\begin{eqnarray*}
\Li_w:w&\mapsto&\Li_w(z)=\sum_{n_1>\ldots>n_r>0}\frac{z^{n_1}}{n_1^{s_1}\ldots n_r^{s_r}},\\[-1mm]
\zeta_w:w&\mapsto&\zeta(w)=\sum_{n_1>\ldots>n_r>0}\frac1{n_1^{s_1}\ldots n_r^{s_r}},\\[-1mm]
\H_w:w&\mapsto&\H_w(N)=\sum_{N\geq n_1>\ldots>n_r>0}\frac1{n_1^{s_1}\ldots n_r^{s_r}},\\[-1mm]
\P_w:w&\mapsto&\P_w(z)=\sum_{N\ge0}\H_w(N)z^N=\frac{\Li_w(z)}{1-z}.
\end{eqnarray*}
Let $\Pi_X:\CYY\longrightarrow\CXX$ and $\Pi_Y:\CXX\longrightarrow\CYY$
denote the ``change" of alphabets over $\CXX$ and $\CYY$ respectively.
\end{slide}

\begin{slide}{Structure of polylogarithms}

Let $\calC=\C[z,z^{-1},(1-z)^{-1}]$
\begin{theorem}[HNM, van der Hoeven \& Petitot, 1998]
Putting $\Li_{x_0}(z)=\log z$,
$\Li:w\mapsto\Li_w$ becomes an isomorphism from $(\C\langle X\rangle,\shuffle)$ to
$(\C\{\Li_w\}_{w\in X^*},.)$.
\end{theorem}
\begin{itemize}
\item $\Li_w,w\in X^*,$ are $\calC$-linearly independent.\\
Then $\{\Li_w\}_{w\in X^*}$ is universal Picard-Vessiot extension
of fuchsian differential equations with three regular singularities.

\item $\Li_l,l\in\Lyn X,$ are $\calC$-algebraically independent.
\item $\zeta(l),l\in\Lyn X\setminus\{x_0,x_1\}$, are generators of the algebra $\calZ$.
\end{itemize}
\end{slide}

\begin{slide}{More about structure of polylogarithms}
{\bf Hoffman} proved that $\forall u,v\in Y^*,\H_u\H_v=\H_{u \stuffle v}$.\\
Therefore, $\forall u,v\in Y^*\setminus y_1Y^*$, $\zeta(u)\zeta(v)=\zeta(u \stuffle v)$.
\begin{eqnarray*}
\P_u(z)\odot\P_v(z)
=\sum_{n\ge0}\H_u(n)\H_v(n)z^n
=\sum_{n\ge0}\H_{u\stuffle v}(n)z^n
=\P_{u\stuffle v}(z).
\end{eqnarray*}
\begin{theorem}[HNM, 2003]
$(\C\{\P_w\}_{w\in Y^*},\odot)\cong(\C\langle Y\rangle,\stuffle)\cong(\C\{\H_w\}_{w\in Y^*},.)$.
\end{theorem}
\begin{itemize}
\item $\P_w,w\in Y^*$, are $\calC$-linearly independent.\\
Then $\H_w,w\in Y^*$, are linearly independent.
\item $\P_l,l\in\Lyn Y$, are $\calC$-algebraically independent.\\
Then $\H_l,l\in\Lyn Y$, algebraically independent.
\item $\zeta(l),l\in\Lyn Y\setminus\{y_1\}$, are generators of the algebra $\calZ$.
\end{itemize}
\end{slide}

\begin{slide}{Towards the structure of polyz\^etas}
\small
\begin{corollary}
$\forall u,v\in X^*,\Li_u\Li_v=\Li_{u\shuffle v}\Rightarrow
\forall u,v\in x_0X^*x_1,\zeta(u)\zeta(v)=\zeta(u\shuffle v)$.
\end{corollary}
\begin{example}
\vskip-13mm
\begin{eqnarray*}
\quad\qquad x_0x_1\shuffle x_0^2x_1&=&x_0x_1x_0^2x_1+3x_0^2x_1x_0x_1+6x_0^3x_1^2,\\
\Li_{2}\Li_{3}&=&\Li_{2,3}+3\Li_{3,2}+6\Li_{4,1},\\
\zeta(2)\zeta(3)&=&\zeta(2,3)+3\zeta(3,2)+6\zeta({4,1}).
\end{eqnarray*}
\end{example}

\begin{corollary}
$\forall u,v\in Y^*,\H_u\H_v=\H_{u \stuffle v}\Rightarrow
\forall u,v\in Y^*\setminus y_1Y^*,\zeta(u)\zeta(v)=\zeta(u\stuffle v)$.
\end{corollary}
\begin{example}
\vskip-13mm
\begin{eqnarray*}
\quad y_{2}\stuffle y_{3}&=&y_{2}y_{3}+y_{3}y_{2}+y_{5},\\
\P_{y_2}\odot\P_{y_3}&=&P_{y_2y_3}+\P_{y_3y_2}+\P_{y_5},\\
\H_{2}\H_{3}&=&\H_{2,3}+\H_{3,2}+\H_{5},\\
\zeta(2)\zeta(3)&=&\zeta(2,3)+\zeta(3,2)+\zeta(5).
\end{eqnarray*}
\end{example}

$$\left.
\begin{array}{l}
\zeta(2)\zeta(3)=\zeta(2,3)+3\zeta(3,2)+6\zeta({4,1})\\
\zeta(2)\zeta(3)=\zeta(2,3)+\zeta(3,2)+\zeta(5)
\end{array}\right\}
\Rightarrow\zeta(5)=2\zeta(3,2)+6\zeta({4,1}).$$
\end{slide}

\begin{slide}{Polynomial relations among $\{\zeta(l)\}_{l\in\Lyn X\setminus\{x_0,x_1\}}$}
\vskip-4mm{\tiny
\begin{eqnarray*}
\zeta(2,1)&=&\zeta(3)\\[-2.5pt]
\zeta(4)&=&\frac25\zeta(2)^2\\[-2.5pt]
\zeta(3,1)&=&\frac1{10}\zeta(2)^2\\[-2.5pt]
\zeta(2,1,1)&=&\frac25\zeta(2)^2\\[-2.5pt]
\zeta(4,1)&=&2\zeta(5)-\zeta(2)\zeta(3)\\[-2.5pt]
\zeta(3,2)&=&-\frac{11}2\zeta(5)+3\zeta(2)\zeta(3)\\[-2.5pt]
\zeta(3,1,1)&=&2\zeta(5)-\zeta(2)\zeta(3)\\[-2.5pt]
\zeta(2,2,1)&=&-\frac{11}2\zeta(5)+3\zeta(2)\zeta(3)\\[-2.5pt]
\zeta(2,1,1,1)&=&\zeta(5)\\[-2.5pt]
\zeta(6)&=&{8\over{35}}{{\zeta(2)}^3}\\[-2.5pt]
\zeta(5,1)&=&-{{1\over2}{{\zeta(3)}^2}}+{{6\over{35}}{{\zeta(2)}^3}}\\[-2.5pt]
\zeta(4,2)&=&{{\zeta(3)}^2} -{{{32}\over{105}}{{\zeta(2)}^3}}\\[-2.5pt]
\zeta(4,1,1)&=&-{{\zeta(3)}^2}+{{{23}\over{70}}{{\zeta(2)}^3}}\\[-2.5pt]
\zeta(3,2,1)&=&{3{{\zeta(3)}^2}} -{{{29}\over{30}}{{\zeta(2)}^3}}\\[-2.5pt]
\zeta(3,1,2)&=&-{{3\over2}{{\zeta(3)}^2}}+{{{53}\over{105}}{{\zeta(2)}^3}}\\[-2.5pt]
\zeta(3,1,1,1)&=&-{{1\over2}{{\zeta(3)}^2}}+{{6\over{35}}{{\zeta(2)}^3}}\\[-2.5pt]
\zeta(2,2,1,1)&=&{{\zeta(3)}^2} -{{{32}\over{105}}{{\zeta(2)}^3}}\\[-2.5pt]
\zeta(2,1,1,1,1)&=&{8\over{35}}{{\zeta(2)}^3}
\end{eqnarray*}
}
\end{slide}

\begin{slide}{Irreducible polyz\^etas by computer}
{\tiny
in Axiom at weight {\bf 10} (avec Petitot),\\
in Maple at weight {\bf 12} (with Bigotte, Jacob, Oussous, Petitot),\\
in C++ at weight {\bf 16} (with El Wardi, Jacob, Oussous, Petitot).\\

$$\begin{tabular}{|c||c|c|c|c|c|}
\hline
$\!\!\displaystyle{\phantom{nn}r\atop n\phantom{rr}}\!\!$
&$\!\!1\!\!$&$\!\!2\!\!$&$\!\!3\!\!$&$\!\!4\!\!$&$\!\!5\!\!$\\[3pt]
\hline
$\!\!2\!\!$&$\!\!\zeta(2)\!\!$&&&&\\[3pt]
\hline
$\!\!3\!\!$&$\!\!\zeta(3)\!\!$&&&&\\[3pt]
\hline
$\!\!5\!\!$&$\!\!\zeta(5)\!\!$&&&&\\[3pt]
\hline
$\!\!7\!\!$&$\!\!\zeta(7)\!\!$&&&&\\[3pt]
\hline
$\!\!8\!\!$&&$\!\!\zeta(6,2)\!\!$&&&\\[3pt]
\hline
$\!\!9\!\!$&$\!\!\zeta(9)\!\!$&&&&\\[3pt]
\hline
$\!\!10\!\!$&&$\!\!\zeta(8,2)\!\!$&&&\\[3pt]
\hline
$\!\!11\!\!$&$\!\!\zeta(11)\!\!$&&$\!\!\zeta(8,2,1)\!\!$&&\\[3pt]
\hline
$\!\!12\!\!$&&$\!\!\zeta(10,2)\!\!$&&$\!\!\zeta(8,2,1,1)\!\!$&\\[3pt]
\hline
$\!\!13\!\!$&$\!\!\zeta(13)\!\!$&&$\!\!\zeta(9,3,1)\!\!$&&\\
&&&$\!\!\zeta10,2,1)\!\!$&&\\[3pt]
\hline
$\!\!14\!\!$&&$\!\!\zeta(10,4)\!\!$&&$\!\!\zeta(10,2,1,1)\!\!$&\\[3pt]
&&$\!\!\zeta(12,2)\!\!$&&&\\[3pt]
\hline
$\!\!15\!\!$&$\!\!\zeta(15)\!\!$&&$\!\!\zeta(11,3,1)\!\!$&&$\!\!\zeta(10,2,1,1,1)\!\!$\\[3pt]
&&&$\!\!\zeta(12,2,1)\!\!$&&\\[3pt]
\hline
$\!\!16\!\!$&&$\!\!\zeta(12,4)\!\!$&&$\!\!\zeta(10,4,1,1)\!\!$&\\[3pt]
&&$\!\!\zeta(14,2)\!\!$&&$\!\!\zeta(11,3,1,1)\!\!$&\\[3pt]
&&&&$\!\!\zeta(12,2,1,1)\!\!$&\\[3pt]
\hline
\end{tabular}$$
}
\end{slide}

\begin{slide}{}
\vfill
\centerline{{\Large NONCOMMUTATIVE GENERATING SERIES}}
\centerline{{\Large TECHNOLOGY}}
\vfill

\end{slide}

\begin{slide}{Noncommutative generating series}
\begin{definition}
$\L(z):=\Sum_{w\in X^*}\Li_{w}(z)\;w\quad$ and $\quad\H(N):=\Sum_{w\in Y^*}\H_w(N)\;w$.
\end{definition}

Let $\Lyn X$ and $l\in\Lyn X$ (resp. $\{\hat l\}_{l\in\Lyn Y}$
and $\{\hat l\}_{l\in\Lyn Y}$) be the transcendence basis of
$(\CX,\shuffle)$ (resp. $(\CY,\stuffle)$) and its dual basis respectively. Then

\begin{theorem}
$\L$ and $\H$ are group-like and\\
$\L(z)=e^{x_1\log\frac1{1-z}}\L_{\reg}(z)e^{x_0\log z}\quad$ and $\quad
\H(N)=e^{\H_1(N)\;y_1}\H_{\reg}(N)$, where
$\ \L_{\reg}(z)=\Prod_{l\in\Lyn X\atop l\neq x_0,x_1}^{\searrow}e^{\Li_l(z)\;\hat l}\quad$
and $\quad\H_{\reg}(N)=\Prod_{l\in\Lyn Y\atop l\neq y_1}^{\nearrow}e^{\H_l(N)\;\hat l}$.
\end{theorem}

\begin{definition}
$Z_{\shuffle}:=\L_{\reg}(1)\quad$ and $\quad Z_{\stuffle}:=\H_{\reg}(\infty)$.
\end{definition}
\end{slide}

\begin{slide}{Double regularization to $\textcolor{red}{0}$}
\begin{proposition}
Let $\zeta_{\minishuffle}:\CXX\longrightarrow\C$
be the shuffle algebra morphism defined by
\begin{itemize}
\item $\zeta_{\minishuffle}(x_0)=\zeta_{\minishuffle}(x_1)=\textcolor{red}{0}$,
\item for any $r_1>1,\zeta_{\minishuffle}(x_0^{r_1-1}x_1\ldots x_0^{r_k-1}x_1)=\zeta(r_1,\ldots,r_k),$
\item for any $u,v\in X^*,
\zeta_{\minishuffle}(u\shuffle v)=\zeta_{\minishuffle}(u)\zeta_{\minishuffle}(v)$.
\end{itemize}
Then $\Sum_{w\in X^*}\zeta_{\minishuffle}(w)\;w=Z_{\shuffle}$.
\end{proposition}

\begin{proposition}
Let $\zeta_{\ministuffle}:\CYY\longrightarrow\C$
be the algebra morphism defined by
\begin{itemize}
\item $\zeta_{\ministuffle}(y_1)=\textcolor{red}{0}$,
\item for any $r_1>1,\zeta_{\ministuffle}(y_{r_1}\ldots y_{r_k})=\zeta(r_1,\ldots,r_k),$
\item for any $u,v\in Y^*,
\zeta_{\minishuffle}(u\stuffle v)=\zeta_{\ministuffle}(u)\zeta_{\ministuffle}(v)$.
\end{itemize}
Then $\Sum_{w\in Y^*}\zeta_{\ministuffle}(w)\;w=Z_{\stuffle}$.
\end{proposition}
\end{slide}

\begin{slide}{Results {\it\`a la} Abel}
\small
\begin{theorem}[HNM, 2005]
%$\L,\H$ and $Z_{\shuffle}$ are group-like and are connected by
\rightline{$\Lim_{z\rightarrow1}e^{y_1\log\frac1{1-z}}\Pi_Y\L(z)=
\Lim_{N\rightarrow\infty}\biggl[\Sum_{k\ge0}\H_{y_1^k}(N)(-y_1)^k\biggr]\H(N)=\Pi_YZ_{\shuffle}$.}
\end{theorem}
\rightline{$\Rightarrow\H(N){}_{\widetilde{N\rightarrow\infty}}
\exp\biggl[-\Sum_{k\ge1}\H_{y_k}(N)\Frac{(-y_1)^k}{k}\biggr]\Pi_YZ_{\shuffle}$.}

\begin{theorem}[Costermans, Enjalbert \& HNM, 2005]
There exists algorithmically computable coefficients $b_i\in\calZ'$,
the $\Q$-algebra generated by $\calZ$ and by $\textcolor{red}{\gamma}$,
$\kappa_i\in\N$ and $\eta_i\in\Z$ such that
$\forall w\in Y^*,\H_w(N){}_{\widetilde{N\rightarrow\infty}}
\Sum_{i=0}^{+\infty}b_iN^{\eta_i}\log^{\kappa_i}(N)$.
\end{theorem}
\vskip-2mm
\begin{example}
\vskip-7mm
\begin{eqnarray*}
\H_{4,2}(N) &=& \zeta (4,2)
-{\frac {{\pi }^{2}}{6}} \,{\displaystyle \frac {1}{N^{3}}}  +
\left({ \frac {{\pi }^{2}}{12}}
+ { \frac {1}{4}} \right){ \frac {  1}{N^{4}}}
-{\left({\frac {{\pi }^{2}}{18}} + {
\frac {2}{5}} \right)\frac {1}{N^{5}}} + \mathrm{O}\left({ \frac
{1}{N^{6}}}\right),\\
\H_{1,4}(N)&=&{\frac {{\pi }^{4}}{90}}\,\ln  \left( N \right)
+{\frac {\,{\pi }^{4}}{90}}\textcolor{red}{\gamma}\,-\zeta \left( 4,1 \right) -
\zeta \left( 5 \right)\\
&+&{\frac {{\pi }^{4}}{180}}\,{\frac {1}{N}}
-{\frac {{\pi }^{4}}{1080}}\,{\frac {1}{{N}^{2}}}+\frac19
\frac1{N^3}+\left(\frac{\pi^4}{10800}-\frac1{24}\right)
\frac1{N^4}+\mathrm{O}\left(\frac1{N^5}\right).
\end{eqnarray*}
\end{example}
\end{slide}

\begin{slide}{Generalized Euler constants}
\small
\begin{theorem}[HNM, 2005]
For any $k\ge0$ and for any $w\in Y^*\setminus\{y_1\}$,
let $\gamma_{y_1^kw}$ be the constant associated to $\H_{y_1^kw}$.
Let $G:=\Sum_{w\in Y^*}\gamma_w\;w$.

Then
$G$ is group-like and $G=\exp\biggl[\textcolor{red}{\gamma}y_1
-\Sum_{k\ge2}\zeta(k)\Frac{(-y_1)^k}{k}\biggr]\Pi_YZ_{\shuffle}$.
\end{theorem}
Let $b_{n,k}(t_1,\ldots,t_k)$ be the Bell polynomials.
By specializing at $t_1=\textcolor{red}{\gamma}$ and for
$l\ge2,t_l=(-1)^{l-1}(l-1)!\zeta(l)$ and by using the identity, for any $u\in X^*$,
$x_1^kx_0u=\Sum_{l=0}^kx_1^l\shuffle(x_0[(-x_1)^{k-l}\shuffle u])$, we get
\begin{corollary}
$\gamma_{y_1^kw}
=\Sum_{i=0}^k\Frac{\zeta(x_0[(-x_1)^{k-i}\minishuffle\Pi_Xw])}{i!}
\biggl[\Sum_{j=1}^ib_{i,j}(\textcolor{red}{\gamma},-\zeta(2),2\zeta(3),\ldots)\biggr]$.\\
In particular,
$\gamma_{y_1^k}=\Sum_{{s_1,\ldots,s_k>0}\atop{s_1+\ldots+ks_k=k}}
\frac{(-1)^k}{s_1!\ldots s_k!}
(-\textcolor{red}{\gamma})^{s_1}\biggl(-\frac{\zeta(2)}{2}\biggr)^{s_2}\ldots
\biggl(-\frac{\zeta(k)}{k}\biggr)^{s_k}$.
\end{corollary}
\end{slide}

\begin{slide}{Regularization to $\textcolor{red}{\gamma}$}
\small
\vskip-5mm
\begin{eqnarray*}
\gamma_{1,1}\!\!\!\!&=&\!\!\!\![\textcolor{red}{\gamma}^2-\zeta(2)]/2,\\
\gamma_{1,1,1}\!\!\!\!&=&\!\!\!\![\textcolor{red}{\gamma}^3-3\zeta(2)\textcolor{red}{\gamma}+2\zeta(3)]/6,\\
\gamma_{1,1,1,1}\!\!\!\!&=&\!\!\!\!
[80\zeta(3)\textcolor{red}{\gamma}-60\zeta(2)\textcolor{red}{\gamma}^2+6\zeta(2)^2+10\textcolor{red}{\gamma}^4]/{240},\\
\gamma_{1,7}\!\!\!\!&=&\!\!\!\!\zeta(7)\textcolor{red}{\gamma}+\zeta(3)\zeta(5)-{54}\zeta(2)^4/{175},\\
\gamma_{1,1,6}\!\!\!\!&=&\!\!\!\!{4}\zeta(2)^3\textcolor{red}{\gamma}^2/{35}+[\zeta(2)\zeta(5)+2\zeta(3)\zeta(2)^2/5-4\zeta(7)]\textcolor{red}{\gamma}\\
&+&\zeta(6,2)+{19}\zeta(2)^4/{35}+\zeta(2)\zeta(3)^2/2-4\zeta(3)\zeta(5),\\
\gamma_{1,1,1,5}\!\!\!\!&=&\!\!\!\!\frac{3\zeta(6,2)}4-\frac{14\zeta(3)\zeta(5)}3+\frac{3\zeta(2)\zeta(3)^2}4+\frac{809\zeta(2)^4}{1400}+\frac{\zeta(5)\textcolor{red}{\gamma}^3}6\\
\!\!\!&+&\!\!\!\biggl[\frac{\zeta(3)^2}4-\frac{\zeta(2)^3}5\biggr]\textcolor{red}{\gamma}^2
-\biggl[2\zeta(7)-\frac{3\zeta(2)\zeta(5)}2+\frac{\zeta(3)\zeta(2)^2}{10}\biggr]\textcolor{red}{\gamma}.
\end{eqnarray*}
\begin{theorem}\label{reg3}
$\gamma_{\bullet}$ realizes the morphism from
$(\CYY,\ministuffle)$ to $(\C,.)$ verifying
\begin{itemize}
\item for any word $u,v\in Y^*,\gamma_{u\ministuffle v}=\gamma_{u}\gamma_{v}$,
\item for any convergent word $w\in Y^*- y_1Y^*,\gamma_{w}=\zeta(w)$,
\item $\gamma_{y_1}=\textcolor{red}{\gamma}$.
\end{itemize}
Then $G=e^{\textcolor{red}{\gamma}y_1}Z_{\ministuffle}$.
\end{theorem}
\end{slide}

\begin{slide}{The meaning of the double regularization to $\textcolor{red}{0}$}
The constant $\gamma_{y_1}=\textcolor{red}{\gamma}$ is obtained as the finite part of
the asymptotic expansion of $\H_1(n)$ in the scale $\{n^{a}\log^{b}(n)\}_{a\in\Z,b\in\N}$.

\vfill

In the same way, since for any $n\in\N$, $n$ and $\H_1(n)$ are algebraically independent
then $\{n^{a}\H_1^{b}(n)\}_{a\in\Z,b\in\N}$ constitutes a new scale for asymptotic expansions.

\vfill

Let $C_1=\Q\oplus x_0\QX x_1$ and $C_2=\Q\oplus(Y\setminus\{y_1\})\QY$.
By the Radford theorem and its generalization over $Y$ (due to Malvenuto \& Reutenauer),
one has respectively
\begin{eqnarray*}
&&(\QX,\shuffle)\cong\Q[\Lyn X]=C_1[x_0,x_1],\\
&&(\QY,\stuffle)\cong\Q[\Lyn Y]=C_2[y_1].
\end{eqnarray*}
Thus, $\zeta_{\minishuffle}(x_1)=\textcolor{red}{0}$
and $\zeta_{\ministuffle}(y_1)=\textcolor{red}{0}$
can be interpreted as the finite part of the asymptotic expansions of
$\Li_1$ and $\H_1$ in the scales $\{(1-z)^{a}\log(1-z)^{b}\}_{a\in\Z,b\in\N}$ 
and $\{n^{a}\H_1^{b}(n)\}_{a\in\Z,b\in\N}$ respectively.

\end{slide}

\begin{slide}{Differential Galois group of polylogarithms}
$\LI_{\calC}$ is the smallest algebra containing $\calC$ 
closed by derivation, by integration w.r.t. $\omega_0$ and $\omega_1$.
It is the $\calC$-modulus generated by $\{\Li_w\}_{w\in X^*}$.

Let $\sigma\in\mathrm{Gal}(\LI_{\calC})$. Then
$\Sum_{w\in X^*}\sigma\Li_w\;w
=\Prod_{l\in\Lyn}^{\searrow}e^{\sigma\Li_{{\check S}_l}S_l}$.\\
Since $d\sigma\Li_{x_i}=\sigma d\Li_{x_i}=\omega_i$
then $\sigma\Li_{x_i}=\Li_{x_i}+c_{x_i}$.\\
More generally,
$\sigma\Li_{{\check S}_l}=\Int\omega_{x_i}
\Frac{\sigma\Li_{{\check S}_{l_1}}^{i_1}}{i_1!}\cdots
\frac{\sigma\Li_{{\check S}_{l_k}}^{i_k}}{i_k!}+c_{{\check S}_l}$.

Consequently,
$\Sum_{w\in X^*}\sigma\Li_w\;w=\L\Prod_{l\in\Lyn}^{\searrow}
e^{c_{{\check S}_l}S_l}=\L e^{C_{\sigma}}.$\\[5pt]
The action of $\sigma\in\mathrm{Gal}(\LI_{\calC})$ over
$\{\Li_w\}_{w\in X^*}$ is equivalent to the action of
$e^{C_{\sigma}}\in\mathrm{Gal}(DE)$ over the exponential solution $\L$. So,

\begin{theorem}[HNM, 2003]
$\mathrm{Gal}(\LI_{\calC})\cong\mathrm{Gal}(DE)=\{e^C\;|\;C\in\LXX\}$.
\end{theorem}
\end{slide}

\begin{slide}{Action of $\mathrm{Gal}(DE)$ on the asymptotic expansions}
\small
\begin{theorem}[Group of associators theorem]
For any commutative $\Q$-algebra $A$, let $\Phi\in\AXX$ and $\Psi\in\AYY$
be group-like elements such that $\Psi=B(y_1)\Pi_Y\Phi$.
There exists an unique $\textcolor{red}{C}\in\LAX$
such that $\Phi=Z_{\shuffle}e^{\textcolor{red}C}$
and $\Psi=G\Pi_Ye^{\textcolor{red}C}$.
\end{theorem}

\vskip3mm

If $\textcolor{red}C\in\LAX$ then $\L'=\L e^{\textcolor{red}C}$
is group-like and $e^{\textcolor{red}C}\in\mathrm{Gal}(DE)$.
Let $\H'(N)$ be the n.c.g.s. of the Taylor coefficients,
belonging the harmonic algebra, of $\{(1-z)^{-1}\L'_w(z)\}_{w\in Y^*}$. 
Then $\H'(N)$ is group-like.

\vskip3mm

\begin{eqnarray*}
\Frac{\L'(1-\eps)}{\eps}{}_{\widetilde{\eps\to 0^+}}e^{-(1+x_1)\log\eps}\Phi_{KZ}e^{\textcolor{red}C}
&\Rightarrow&
\H'(N){}_{\widetilde{N\to\infty}}\H(N)\Pi_Ye^{\textcolor{red}C}.
\end{eqnarray*}
Let $\kappa_w$ be the constant part of $\H'_w(N)$. Then,
\begin{eqnarray*}
\sum_{w\in Y^*}\kappa_w\;w=\Psi_{KZ}\Pi_Ye^{\textcolor{red}C},
&\mbox{or eqivalenty}&
\Pi_X\sum_{w\in Y^*}\kappa_w\;w=B^{-1}(x_1)\Phi_{KZ}e^{\textcolor{red}C}.
\end{eqnarray*}
We put then $\Psi:=Z_{\shuffle}\Pi_Ye^{\textcolor{red}C}$
and $\Phi:=G\Pi_Ye^{\textcolor{red}C}$.
\end{slide}

\begin{slide}{Examples (action of the monodromy group)}
For $t\in]0,1[$, the monodromies around $0,1$ of $\L$
are given respectively by ($\textcolor{red}{p}=2i\pi$)
\begin{eqnarray*}
\calM_0\L(t)=\L(t)e^{\textcolor{red}{p}\mathfrak{m}_0}&\mbox{and}&
\calM_1\L(t)=\L(t)\Phi_{KZ}^{-1}e^{-\textcolor{red}{p}x_1}\Phi_{KZ}\\
&&\phantom{\calM_1\L(t)}=\L(t)e^{\textcolor{red}{p}\mathfrak{m}_1},\\
\mbox{where}\quad
\mathfrak{m}_0=x_0&\mbox{and}&
\mathfrak{m}_1=\Prod_{l\in\Lyn,l\neq x_0,x_1}^{\searrow}
e^{-\zeta(\check S_l)\ad_{S_l}}(-x_1).
\end{eqnarray*}

\begin{itemize}

\item If $\textcolor{red}C=\textcolor{red}{p}\mathfrak{m}_0$ then
$\Phi=Z_{\shuffle}e^{\textcolor{red}{p}x_0}$ and
$\Psi=\exp\biggl[\textcolor{red}{\gamma}y_1
-\Sum_{k\ge2}\zeta(k)\Frac{(-y_1)^k}{k}\biggr]\Pi_YZ_{\shuffle}=G$.

\item If $\textcolor{red}C=\textcolor{red}{p}\mathfrak{m}_1$ then
$\Phi=e^{-\textcolor{red}{p}x_1}Z_{\shuffle}$ and
$\Psi=\exp\biggl[(\underbrace{\textcolor{red}{\gamma-p}}_{=\textcolor{red}T})y_1
-\Sum_{k\ge2}\zeta(k)\Frac{(-y_1)^k}{k}\biggr]\Pi_YZ_{\shuffle}
=e^{-\textcolor{red}{p}y_1}G$.
\end{itemize}
Hence, the monodromies could not allow neither to introduce the factor
$e^{\gamma x_1}$
on the left of $Z_{\shuffle}$, neither to eliminate the left factor
$e^{\gamma y_1}$ in $G$ (by putting $T=0$, for example).
\end{slide}

\begin{slide}{Polynomial relations among generators of polyz\^etas}

$\{\zeta_{\minishuffle}(l)\}_{l\in\Lyn X}$ and $\{\zeta_{\ministuffle}(l)\}_{l\in\Lyn Y}$
are also generators of the algebras $\calZ$ and $\calZ'$. 
One also gets
\begin{theorem}
\textcolor{red}{$\Prod_{l\in\Lyn X,l\neq x_0,x_1}^{\searrow}e^{\zeta(l)\;\hat l}
=e^{\Sum_{k\ge2}\zeta(k)\Frac{(-x_1)^k}{k}}
\Pi_X\Prod_{l\in\Lyn Y,l\neq y_1}^{\nearrow}e^{\zeta(l)\;\hat l}$.}
\end{theorem}

Since $\forall l\in\Lyn Y\iff\Pi_Xl\in\Lyn X\setminus\{x_0\}$ then
identifying the local coordinates, in the Lyndon-PBW basis,
we get polynomial relations among these generators
which are algebraically independent on $\textcolor{red}{\gamma}$.

\begin{corollary}
For any $l\in\Lyn Y\setminus\{y_1\}$, let $P_l$ be
the decomposition of $\Pi_X\hat l$ in the Lyndon-PBW basis, over $X$,
and let $\check P_l$ be its dual. Then \textcolor{red}{$\Pi_Xl-\check P_l\in\ker\zeta$}.
Moreover, $\Pi_X\lambda-\check P_\lambda$ is \textcolor{red}{homogenous} of degree $|\lambda|$
and if $\Pi_Xl=\check P_l$ then $\zeta(l)$ is \textcolor{red}{irreducible}.
\end{corollary}
\end{slide}

\begin{slide}{Structure of polyz\^etas}
\begin{theorem}[Structure of polyz\^etas]
The $\Q$-algebra generated by convergent polyz\^etas, $\calZ$,
is isomorphic to the graded algebra $(\Q\oplus(Y-y_1)\QY/\ker\zeta,\stuffle)$.
\end{theorem}

\begin{proof}
Since $\ker\zeta$ is an ideal generated by the homogenous polynomials
then the quotient $\Q\oplus(Y-y_1)\QY/\ker\zeta$ is graded.
\end{proof}

\begin{corollary}
The $\Q$-algebra of polyz\^etas is freely generated by irreducible polyz\^etas.
\end{corollary}
\begin{proof}
For any $\lambda\in\Lyn Y$, if $\lambda=\check P_\lambda$ then one gets the conclusion else
$\Pi_X\lambda-\check P_\lambda\in\ker\zeta$.
Since $\check P_\lambda\in\Q[\Lyn X]$ then $\check P_\lambda$ is polynomial
on Lyndon words of degree $\le|\lambda|$. For each Lyndon word
does appear in this decomposition of $\check P_\lambda$, after applying $\Pi_Y$,
the same process goes on untill having irreducible polyz\^etas.
\end{proof}
\end{slide}

\begin{slide}{Towards the transcendence of $\textcolor{red}{\gamma}$ over $\calZ$}
By considering the commutative indeterminates $t_1,t_2,\ldots$,
then let $A=\Q[t_1,t_2,\ldots]$.

\begin{lemma}
For any $\Phi\in\{\Phi_{KZ}e^{\textcolor{red}C}|\textcolor{red}C\in\LAX\}$,
one get
\begin{eqnarray*}
\Psi=B(y_1)\Pi_Y\Phi&\iff&\Psi'=B'(y_1)\Pi_Y\Phi.
\end{eqnarray*}
\end{lemma}

\begin{theorem}
For \textcolor{red}{all} $\Phi\in\{\Phi_{KZ}e^{\textcolor{red}C}|\textcolor{red}C\in\LQX\}$,
the identies $\Psi=B(y_1)\Pi_Y\Phi$ yield {\textcolor{red}{all}} polynomial relations
among convergent polyz\^etas.
\end{theorem}

Moreover, these relations are algebraically independent on $\textcolor{red}{\gamma}$.
In other words, under the hypothesis $\textcolor{red}{\gamma}\notin\Q$,
the constant $\textcolor{red}{\gamma}$ 
does not verify any polynomial of coefficients in $\calZ$.

\begin{corollary}
If $\textcolor{red}{\gamma}\notin\Q$ then it is transcendental over $\calZ$.
\end{corollary}
\end{slide}

\begin{slide}{}
\vfill
\centerline{{\Large COLOURED POLYLOGARITHMS,}}
\centerline{{\Large HARMONIC SUMS AND POLYZETAS}}
\vfill

\end{slide}

\begin{slide}{Sommes harmoniques et polylogarithmes color\'es}

Let$$\left({\mathbf{b}\atop\mathbf{s}}\right)=\left(b_1,\ldots,b_r\atop s_1,\ldots,s_r\right),$$
where $b_1,..,b_r\in\{1,q,\ldots,q^{n-1}\}$ and $q=e^{2\mathrm{i}\pi/n}$.
\begin{eqnarray*}
\H_{\mathbf{b}\atop\mathbf{s}}(N)&=&\sum_{N\geq n_1>\ldots>n_r>0}
\frac{b_1^{n_1}\cdots b_r^{n_r}}{n_1^{s_1}\ldots n_r^{s_r}},\\
\Li_{\mathbf{b}\atop\mathbf{s}}(z)&=&\sum_{n_1>\cdots>n_r>0}
\frac{b_1^{n_1}\cdots b_r^{n_r}}{n_1^{s_1}\cdots n_r^{s_r}}z^{n_1}.
\end{eqnarray*}

If $\left({b_1\atop s_1}\right)\neq\left({1\atop1}\right)$ then
\begin{eqnarray*}
\lim_{N \rightarrow \infty}\H_{\mathbf{b}\atop\mathbf{s}}(N)
=\lim_{z \rightarrow 1}\Li_{\mathbf{b}\atop\mathbf{s}}(z)
=\sum_{n_1>\ldots>n_r>0}
\frac{b_1^{n_1}\cdots b_r^{n_r}}{n_1^{s_1}\ldots n_r^{s_r}}
=\zeta\left(\mathbf{b}\atop\mathbf{s}\right).
\end{eqnarray*}
\begin{eqnarray*}
\P_{\mathbf{b}\atop\mathbf{s}}(z)=\sum_{n\ge0}\H_{\mathbf{b}\atop\mathbf{s}}(n)z^n
=\frac{\Li_{\mathbf{b}\atop\mathbf{s}}(z)}{1-z}.
\end{eqnarray*}
\end{slide}

\begin{slide}{Coloured alphabets}

Let us consider the alphabets $X=\{x_0,x_q,\ldots,x_{q^n}\}$ and $Y=\{y_{b\atop k}\}_{b\in\{1,\ldots,q^{n-1}\}\atop k\in\N_+}$.
$$\left(b_1,\ldots,b_r\atop s_1,\ldots,s_r\right)\leftrightarrow y_{b_1\atop s_1}\ldots y_{b_r\atop s_r}
\leftrightarrow x_0^{s_1-1}x_{\tau_1}\cdots x_0^{s_r-1}x_{\tau_r},$$
where $j=1,..,r$, $\tau_j=\Prod_{i=1}^{j}b_i$.

\begin{example}
$\left(-1,1,-1\atop\phantom{-}3,4,\phantom{-}5\right)\leftrightarrow x_0^2x_{-1}x_0^3x_{-1}x_0^4x_1.$
\end{example}

Let $\calZ$ be the $\Q$-algebra generated by convergent coloured polyz\^etas.\\
Let $\calZ'$ be the $\Q[\textcolor{red}{\gamma}]$-algebra generated by convergent coloured polyz\^etas.
\end{slide}

\begin{slide}{Harmonic algebra}

For $u,v \in Y^*$, we have $\H_u\H_v=\H_{u \star v},$ where $u\star v$ is defined by $\eps\star u=u\star\eps=u$ and $(y_{b\atop i}u)\star(y_{c\atop j}v)=y_{b\atop i}(u\star y_{c\atop j}v)+y_{c\atop j}(y_{b\atop i}u\star v)+y_{bc\atop i+j}(u\star v)$.\\
By consequent, $\P_u\odot\P_v=\P_{u \star v}.$

\begin{example}\vskip-8mm
{\small
\begin{eqnarray*}
y_{-1\atop 2}\star y_{-1\atop 3}
&=&y_{-1\atop 2}y_{-1\atop 3}+y_{-1\atop 3}y_{-1\atop 2}+y_{1\atop 5},\\
\H_{-1\atop 2}\H_{-1\atop 3}
&=&\H_{-1,-1\atop 2,3}+\H_{-1,-1\atop 3,2}+\H_{1\atop 5},\\
\P_{-1\atop 2}\odot\P_{-1\atop 3}
&=&\P_{-1,-1\atop 2,3}+\P_{-1,-1\atop 3,2}+\P_{1\atop 5},\\
\\
y_{-1\atop 2}y_{-1\atop 5}\star y_{1\atop 4}&=&y_{-1\atop 2}(y_{-1\atop 5}\star y_{1\atop 4})+y_{1\atop 4}(y_{-1\atop 2}
y_{-1\atop 5}\star\eps)+y_{-1\atop 6}(y_{-1\atop 5}\star\eps)\\
&=&y_{-1\atop2}(y_{-1\atop5}y_{1\atop4}+y_{1\atop4}y_{-1\atop5}+y_{-1\atop9})+y_{1\atop4}y_{-1\atop2}y_{-1\atop5}
+ y_{-1\atop6}y_{-1\atop5}\\
&=&y_{-1\atop 2}y_{-1\atop 5}y_{1\atop 4}+y_{-1\atop 2}y_{1\atop 4}y_{-1\atop 5}+y_{1\atop 4}y_{-1\atop 2}
y_{-1\atop 5}+y_{-1\atop 2}y_{-1\atop 9}+ y_{-1\atop 6}y_{-1\atop 5},\\
\H_{-1,-1\atop 2,5}\H_{1\atop 4}
&=&\H_{-1,-1,1\atop 2,5,4}
+\H_{-1,1,-1\atop 2,4,5}+\H_{1,-1,-1\atop 4,2,5}
+\H_{-1,-1\atop 2,9}+\H_{-1,-1\atop 6,5},\\
\P_{-1,-1\atop 2,5}\odot\P_{1\atop 4}
&=&\P_{-1,-1,1\atop 2,5,4}
+\P_{-1,1,-1\atop 2,4,5}+\P_{1,-1,-1\atop 4,2,5}
+\P_{-1,-1\atop 2,9}+\P_{-1,-1\atop 6,5}.
\end{eqnarray*}
}
\end{example}
\end{slide}

\begin{slide}{Structure theorems}
Let $\calO=\{0\}\cup\{q^n=1\}$ and let $\calC=\C[z,\{a_i(z)\}_{i\in\calO}]$
where $a_0(z)=z^{-1}$ and, for $\rho\in\calO\setminus\{0\}$, $a_{\rho}(z)=\rho(1-{\rho}z)^{-1}$.

\begin{theorem}[Bigotte, Jacob, Oussous, Petitot, 2000]
$(\calC\{\Li_w\}_{w\in X^*},.)\simeq(\calC[\Li_{\Lyn X}],.)$.
\end{theorem}

\begin{prop}
$(\C\{\P_w\}_{w\in Y^*},\odot)\simeq(\C\{\H_w\}_{w \in Y^*},.)\simeq(\C\langle Y\rangle,\star)$.
\end{prop}

\begin{theorem}[HNM, 2003]
$(\C\{\P_w\}_{w\in Y^*},\odot)\simeq(\C[\P_{\Lyn Y}],\odot)$,\\
$(\C\{\H_w\}_{w\in Y^*},.)\simeq(\C[\H_{\Lyn Y}],.)$.
\end{theorem}


\end{slide}

\begin{slide}{Distribution relation}

\begin{prop}\label{pro:binomial}
Let $q$ be a $m$-th root of unity. Let $\mathbf s$ and $\mathbf i$ be two compositions.\\
Assume that $0\le i_1<\ldots<i_r\le m-1$. Then,
\begin{eqnarray*}
\zeta\binom{\mathbf s}{q^{\mathbf i}}=\dfrac1{m^r}\sum_{b_1,\ldots,b_r=1}^{m}\sum_{j_1,\ldots,j_r=0}^{m-1}\dfrac{q^{\sum_{l=1}^r(i_1+\ldots+i_l)b_l}}{q^{\sum_{l=1}^r(j_1+\ldots+j_l)b_l}}\zeta\binom{\mathbf s}{q^{\mathbf j}},
\end{eqnarray*}
or equivalently
\begin{eqnarray*}
\sum_{b_1,\ldots,b_r=1}^{m}
\sum_{\stackrel{j_1,\ldots,j_r=0}{j_1\ne i_1,\ldots,j_r\ne i_r}}^{m-1}
\dfrac{\prod_{l=1}^rq^{\sum_{l=1}^r}(i_1+\ldots+i_l)b_l}{q^{\sum_{l=1}^r}(j_1+\ldots+j_l)b_l}
\zeta\binom{\mathbf s}{q^{\mathbf j}}
=0.
\end{eqnarray*}
\end{prop}
\end{slide}

\begin{slide}{Asymptotic expansions}
\begin{theorem}[HNM, 2005]
$(\calC\{\P_w\}_{w\in Y^*},\odot)\simeq(\calC[\P_{Lyn(Y)}],\odot)$ and for any $g\in\calC\{\P_w\}_{w \in Y^*}$,
there exists algorithmically computable coefficients $b_i\in\C$,
the $\Q$-algebra generated by $\calZ$ and by $\textcolor{red}{\gamma}$,
$\kappa_i\in\N$ and $\eta_i\in\Z$ such that
$g(z)\sim\Sum_{j=0}^{+\infty}c_j(1-z)^{\alpha_j}\P_{y_1^{\beta_j}}(z)$ for $z\rightarrow1$.\\
By conssequence, there exists algorithmically computable coefficients $b_i\in\C,\eta_i\in\Z$ and $\kappa_i\in\N$ such that $[z^n]g(z)\sim\Sum_{i=0}^{+\infty}b_in^{\eta_i}\log^{\kappa_i}(n)$ for $n\rightarrow\infty$.
\end{theorem}

\begin{corollary}
There exists algorithmically computable coefficients $b_i\in\calZ',\eta_i\in\Z$ and $\kappa_i\in\N$ such that\\
$\forall w\in Y^*,\H_w(N)\sim\Sum_{i=0}^{+\infty}b_iN^{\eta_i}\log^{\kappa_i}(N)$ for $N\rightarrow+\infty$.
\end{corollary}
\end{slide}

\begin{slide}{}
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\centerline{{\Large THANK YOU FOR YOUR ATTENTION}}
\vfill

\end{slide}

\end{document}
