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\title{{\bf Combinatorial Physics:\\ 
Sch\"utzenberger Factorization and non-commutative differential equations.}}

\author{G H E Duchamp$^{\dag}$,\\ 
 P Blasiak$^{\ddag}$, A Horzela$^\ddag$, Hoang Ngoc Minh$^{\dag}$,\\ K A Penson$^{\diamondsuit}$ and A I Solomon$^{\diamondsuit,\sharp}$}
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\address{\rm
$^a$ LIPN - UMR 7030\\
CNRS - Universit\'e Paris 13\\
F-93430 Villetaneuse, France}

\address{\rm
$^\ddag$ H.Niewodnicza\'nski Institute of
Nuclear Physics, Polish Academy of Sciences\\
ul. Eliasza-Radzikowskiego 152,  PL 31342 Krak\'ow, Poland}

\address{\rm
$^\diamondsuit$ Laboratoire de Physique Th\'eorique de la Mat\`ere Condens\'ee,\\
Universit\'e Pierre et Marie Curie, CNRS UMR 7600\\
Tour 24 - 2e \'et., 4 pl. Jussieu, F 75252 Paris Cedex 05, France}

\address{\rm
$^{\sharp}$ The Open University, Physics and Astronomy Department\\
Milton Keynes MK7 6AA, United Kingdom}

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\maketitle

%\begin{abstract}
{\bf Abstract}\\
The original Hopf algebra inherent to the pre-Feynman diagrams (LDIAG) of the 
Quantum Field Theory of Partitions (QFTP) admits a three-parameter deformation. 
It turns out that the two first parameters (deformation of the structure of algebra) 
are of different natures : the first is due to the deformation of the (tensor) space 
by a commutation factor and the second to a perturbation of the comultiplication. 
Details for these tools used for deformation will be indicated (coloured products, 
dual laws, diagonal deformation, Sweedler's duals).\\ 
As a byproduct, one gets an unexpected deformation of the Euler-Zagier sums. 
It will be shown also how these sums arise from a non-commutative fuchsian differential equation which produces a group-like element. We indicate a way of regularizing them through Sch\"utzenberger factorization. The talk is based on the two papers \cite{DTPK,DMSG}.
%\end{abstract}

\begin{thebibliography}{ABC}
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\bibitem{DTPK} G. H. E. Duchamp, C. Tollu, K. A. Penson, G. Koshevoy, {\it Combinatorial Deformations of Algebras: Twisting and Perturbations}, S�minaire Lotharingien de Combinatoire, to appear.\\
{\tt arXiv:0903.2101v2}
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\bibitem{DMSG} G. H.E. Duchamp,  Hoang Ngoc Minh, Allan I. Solomon, Silvia Goodenough, {\it An interface between physics and number theory}, Proceedings of Group 28 (Group-Theoretical methods in Physics), Newcastle 2010, to appear.\\
{\tt arXiv:1011.0523v1}
\end{thebibliography}
\end{document}