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Intégrales Itérées en Physique Combinatoire

Matthieu Deneufchâtel 1
1 CALIN
LIPN - Laboratoire d'Informatique de Paris-Nord
Abstract : We present several results linked by the tools and by the underlying structures we use (iterated integrals, shuffle products). In the first part, we are interested in the computation of integrals of Selberg type and in their asymptotics when the number of variables tends to infinity. In the general case, we show that the result can be expressed as a product whose number of factors does not depend on the number of variables (under certain conditions). If the power of the Vandermonde determinant equals 2, the limit of the integral when the number of variables tends to infinity can be computed with operators related to Newton's interpolation. The second part has two sections which are related to special functions called hyperlogarithms.\\ We start with the question of the linear independence of a family of functions obtained by iterated integrals and give a criterion that links the properties of the whole family to the behavior of the functions obtained by simple integrals. We show how to construct the required fields of germs of analytic functions which play an important role. Several examples allow us to extend the known results. Then we come back to the free algebra and the properties of dual families, our main interest being Schützenberger's factorisation. We recall some classical results in the partially commutative case ; then we consider the family obtained by dualisation from the Lyndon words. It is not possible to write the factorisation for these dual families but we make precise the nature of the elements of the family obtained by duality. Finally, we present a criterion that gives a condition on the dual families for the factorisation to hold.
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Submitted on : Monday, October 1, 2012 - 11:02:58 AM
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Matthieu Deneufchâtel. Intégrales Itérées en Physique Combinatoire. Combinatoire [math.CO]. Université Paris-Nord - Paris XIII, 2012. Français. ⟨tel-00736727⟩

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