XXII WORKSHOP ON GEOMETRIC METHODS IN PHYSICS
29 JUNE - 5 JULY 2003, BIA£OWIE¯A, POLAND
Karol A.Penson -
Extending Dobinski relations:
from boson normal ordering to Feynman diagrams
We define and investigate the sequences of integers appearing in
in the solution of the normal ordering problem of the n-th power
of a monomial (a+)^r*a^s where a+ and a are boson creation and
annihilation operators satisfying [a,a+]=1, with n,r,s integers.
These sequences are generalizations of conventional (r=s=1) Bell
and Stirling numbers. A comprehensive study of these combinatorial
numbers yields the generating functions, recursion relations and
closed-form expressions as the infinite series. These last forms
extend the celebrated Dobinski relations satisfied by the
conventional Bell numbers [1]. We demonstrate that the generalized
Bell numbers are moments of positive measures on a positive half-
axis. We solve the corresponding Stieltjes moment problem and
obtain explicit forms for the measures by means of the inverse
Mellin transform technique [2].
The properties of these sequences are incorporated in the forma-
lism of zero-dimensional Quantum Field Theory, in the context
of a graphical representation of its perturbative series (Feynman
diagrams) and the exponential formula [3]. We shall exemplify
several exactly soluble situations in which the terms of Feynman
diagrams expansion can be enumerated. They include Field Theories
described by constrained partitions, involutions, generalized
idempotent numbers and their multisections. Many properties of
such expansions can be conveniently described in terms of Hermite-
Kampé de Fériet polynomials [4].
[1]. P.Blasiak, K.A.Penson and A.I.Solomon,"The general boson
normal ordering problem", Phys.Lett.A 309 (2003),198.
[2]. P.Blasiak, K.A.Penson and A.I.Solomon,"Dobinski-type relations
and the log-normal distribution", J.Phys.A (Lett.) 36,(2003),
L273.
[3]. C.M.Bender, D.C.Brody and B.K.Meister,"Quantum field theory
of partitions", J.Math.Phys.40 (1999) 3239.
[4]. P.Appell and J.Kampé de Fériet,"Fonctions Hypergéométriques
et Hypersphériques. Polynomes d'Hermite" (Gauthiers-Villars,
Paris,1926).