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Autour des surpartitions et des identités de type Rogers-Ramanujan

Abstract : A partition of a nonnegative integer is a way of writing this number as a sum of positive integers where order does not matter. Several generalizations of partitions have been studied, among which overpartitions, which are partitions where the last occurrence of a number can be overlined, overpartition pairs, and n-color partitions, which are related to a model of statistical physics. In this thesis, we generalize to overpartition pairs the Andrews-Gordon identities, which are an extension of a classical result of partition theory : the Rogers-Ramanujan identities. To do this, we define two classes basic hypergeometric series and we show that they are generating functions for overpartition pairs satisfying various kinds of conditions (multiplicities, successive ranks, Durfee dissection) and for certain lattice paths. We also show that for some values of the parameters, these series can be written as infinite products, which leads to several Rogers-Ramanujan-type-identities. The proof uses various combinatorial and analytical methods. Finally, we define a generalization of n-color partitions, called n-color overpartitions, and we use these objects to interpret combinatorially certain multiple series and prove other Rogers-Ramanujan-type-identities.
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https://tel.archives-ouvertes.fr/tel-00366067
Contributor : Olivier Mallet <>
Submitted on : Thursday, March 5, 2009 - 4:11:34 PM
Last modification on : Saturday, March 28, 2020 - 2:08:07 AM
Long-term archiving on: : Tuesday, June 8, 2010 - 11:09:43 PM

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  • HAL Id : tel-00366067, version 1

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Olivier Mallet. Autour des surpartitions et des identités de type Rogers-Ramanujan. Autre [cs.OH]. Université Paris-Diderot - Paris VII, 2008. Français. ⟨tel-00366067⟩

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