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arXiv:2606.16647

Generalized Bell polynomial operators arising from generalized normal ordering

Taekyun Kim, Dae Young Kim

Published Jun 15, 2026• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2606.16647

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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 Nfn=(\fa)nN_f^n = (\fa)^nNfn​=(\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.
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.
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 Dobihski 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.
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.
We resolve the ladder operators normal ordering problem for strings in the form for ( q , p )-deformed supersymmetric and shape-invariant potential systems, where n is a positive integer. We provide exact and explicit expressions for their normal form , where in all are at the right-hand side, and find that the solution involves expansion-coefficients sequence which corresponds to the generalization, for any ( q , p )-deformed shape-invariant potential system, of the classical Stirling and Bell numbers. We show that these numbers are given for expressions depending on parameters related to the forms of supersymmetric partner potentials and the system quantum deformation. We apply the general formalism to Pöschl–Teller, Morse and Scarf potential systems.
We resolve the normal ordering problem for symmetric (Dˆ+Dˆ−)n and asymmetric (Dˆ+rDˆ−)n strings of the nonlinear deformed ladder operators Dˆ± for supersymmetric and shape-invariant potential systems, where r and n are positive integers. We provide exact and explicit expressions for their normal forms N{(Dˆ+Dˆ−)n} and N{(Dˆ+rDˆ−)n}, where in N{...} all Dˆ− are at the right side. We find that the solutions involve sequence of expansion coefficients which, for r, n > 1, corresponds to the f-deformed generalization of the classical Stirling and Bell numbers of the second kind. We apply the general formalism for the translational shape-invariant potential systems as well as for the particular case of the harmonic oscillator potential system. We show that these numbers are obtained for families of polynomial expressions generated with the deformations parameters and the parameters related to the forms of the supersymmetric partner potentials.