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Title: Some Combinatorial Problems Inherent in and Related to Quantum Statistics
Speaker: K. A. Penson ( LPTMC, Université de Paris VI)
We shall present a general view of combinatorial aspects of the normal ordering of functions of Boson
creation and annihilation operators. It will be shown that this procedure naturally leads to far reaching
generalizations of classical combinatorial Bell and Stirling numbers for which closed form analytic
formulas in terms of hypergeometric functions are obtained. These representations, termed extended
Dobinski-type relations, allow to consider all these sequences as Stieltjes moments of probability
distributions whose explicit forms are obtained via inverse Mellin transform.
Links with the forthcoming presentations by Blasiak and by Duchamp will be pointed out.
Title: Graphs for Quantum Boson problems
Speaker: P. Blasiak (Institute of Nuclear Physics, Polish Academy of
Sciences, Krakow)
Algebraic structure of Quantum Boson systems is considered from the combinatorial point of view. It is
shown that by lifting to the richer algebra of graphs, operator calculus gains simple interpretation as
the shadow of natural operations on graphs. This provides insights into the algebraic structure of
the theory and sheds light on the combinatorial nature hidden behind its formalism. Practical utility of
this approach is illustrated on examples resolved by methods of symbolic combinatorics.
Title: Feynman-like combinatorial diagrams and the EGF Hadamard Product
Speaker: G.H.E.Duchamp (LIPN, Université Paris XIII)
We consider two aspects of the product formula for formal power series applied to combinatorial field
theories. Firstly, we remark that the case when the functions involved in the product formula are free
exponentials (like in the derivation of Bell polynomials) is of special interest as it leads to groups of
substitutions and a correspondence with vector fields on the line. Secondly, we discuss deformations
(counting natural graph parameters as crossings and superpositions) of the Feynman-like algebra
arising from the product formula of two free exponentials. This results in a true Hopf deformation of
this algebra.