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International Journal of Geometric Methods in Modern Physics

On generalized Bell numbers and normal ordering problem

A. Ouahhabi, E. H. Tahri

Published Jul 30, 2025• 0 citations
Publisher

Abstract

In this paper, the general framework to construct generalized coherent states for the generalized Heisenberg algebra, using a [Formula: see text]-analog of exponential function and [Formula: see text]-integers, is discussed. Based on these notions, generalized Bell polynomials and Bell numbers, and so their exponential generating functions as well as a [Formula: see text]-analog of the latter and generalized Dobiński formulas are obtained. Then, the normal ordered form of exponential function of generalized number operator and its [Formula: see text]-analog exponential are investigated for general characteristic functions. In particular, the Arik–Coon [Formula: see text]-deformed quantum harmonic oscillator and the undeformed quantum physical system of infinite square well potential cases are well studied.

DOI
10.1142/s0219887825502858

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This paper explores the deformed combinatorial structures arising from the generalized Heisenberg algebra GHA which is characterized by an analytic function of the Hamiltonian f(H) and governs systems with non-linear spectra. Moving beyond the classical Heisenberg-Weyl framework, we investigate the normal ordering of the generalized number operator $N_f^n = (\fa)^n$, which naturally introduces the generalized Stirling operators of the second kind. Using the vacuum eigenvalue the Hamiltonian, we define quantum operator factorials and a generalized quantum exponential function. We explicitly construct the generalized coherent states and derive several operator identities. Notably, we prove that the powers of quantum operators expand into generalized falling factorials, and that the coherent state expectation values are explicitly given by the generalized Bell polynomial operators.
This paper introduces a unified operational framework that bridges combinatorial mathematics and nonlinear quantum mechanics by systematically developing the theory of generalized, degenerate, and rrr -extended operator-valued Stirling numbers and Bell polynomials. Constructed within the framework of the Generalized Heisenberg Algebra GHAf\text{GHA}_fGHAf​ , where the standard commutation relation is modulated by an analytic function of the Hamiltonian f(H)f(H)f(H) , these novel algebraic objects capture the intrinsic nonlinearity of deformed quantum systems. By implementing operational methods and evaluating operator actions on coherent and number states, we establish explicit representations, structural recurrence relations, and robust generating functions. We demonstrate that in the special case of f(H)=H+1f(H) = H + 1f(H)=H+1 , our operator-valued formulations gracefully reduce to their classical combinatorial counterparts, successfully recovering foundational milestones such as Dobinski’s formula.
We derive explicit formulas for the normal ordering of powers of arbitrary monomials of boson operators. These formulas lead to generalisations of conventional Bell and Stirling numbers and to appropriate generalisations of the Dobinski relations. These new combinatorial numbers are shown to be coherent state matrix elements of powers of the monomials in question. It is further demonstrated that such Bell-type numbers, when considered as power moments, give rise to positive measures on the positive half-axis, which in many cases can be written in terms of known functions.
The Heisenberg algebra is first deformed with the set of parameters q,l,λ{q, l, λ}q,l,λ to generate a new family of generalized coherent states. In this framework, the matrix elements of relevant operators are exactly computed. A proof on sub-Poissonian character of the statistics of the main deformed states is provided. This property is used to determine a generalized metric. A unified method of calculating structure functions from commutation relations of deformed single-mode oscillator algebras is then presented. A natural approach to building coherent states associated to deformed algebras is deduced. Known deformed algebras are given as illustration. Futhermore, we generalize a class of two-parameter deformed Heisenberg algebras related to meromorphic functions, called R(p,q){\cal R}(p,q)R(p,q)-deformed algebra. Relevant families of coherent states maps are probed and their corresponding hypergeometric series are computed. The latter generalizes known hypergeometric series and gives to a generalization of the binomial theorem. The involved notions of differentiation and integration generalize the usual qqq- and (p,q)(p,q)(p,q)-differentiation and integration. A Hopf algebra structure compatible with the R(p,q){\cal R}(p,q)R(p,q)-algebra is deduced. We succeed in giving a new characterization of Rogers- Szegö polynomials, called R(p,q){\cal R}(p,q)R(p,q)-deformed Rogers-Szegö polynomials, by their three-term recursion relations and the associated quantum algebra built with corresponding creation and annihilation operators. Continuous R(p,q){\cal R}(p,q)R(p,q)-deformed Hermite polynomials and their recursion relation are also deduced. Novel algebraic relations are provided and discussed. The whole formalism is performed in a unified way, generalizing known relevant results which are straightforwardly derived as particular cases.
We investigate properties of exponential operators preserving the particle number using combinatorial methods developed in order to solve the boson normal ordering problem. In particular, we apply generalized Dobinski relations and methods of multivariate Bell polynomials which enable us to understand the meaning of perturbation-like expansions of exponential operators. Such expansions, obtained as formal power series, are everywhere divergent but the Pade summation method is shown to give results which very well agree with exact solutions got for simplified quantum models of the one mode bosonic systems.