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).