Generalized Bell polynomial operators arising from generalized normal ordering

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.