Combinatorial approach to generalized Bell and Stirling numbers and boson normal ordering problem

Abstract

We consider the numbers arising in the problem of normal ordering of expressions in canonical boson creation and annihilation operators. We treat a general form of a boson string which is shown to be associated with generalizations of Stirling and Bell numbers. The recurrence relations and closed-form expressions (Dobiski-type formulas) are obtained for these quantities by both algebraic and combinatorial methods. By extensive use of methods of combinatorial analysis we prove the equivalence of the aforementioned problem to the enumeration of special families of graphs. This link provides a combinatorial interpretation of the numbers arising in this normal ordering problem.

10 pages, 5 figures

References in corpus (5)

• Normal Ordering for Deformed Boson Operators and Operator-valued Deformed Stirling Numbers
• The general boson normal ordering problem
• Boson Normal Ordering via Substitutions and Sheffer-type Polynomials
• Some useful combinatorial formulae for bosonic operators
• Quadratic integrals of motion for the systems of identical particles

Cited by in corpus (12)

• Combinatorics and Boson normal ordering: A gentle introduction
• Laguerre-type derivatives: Dobinski relations and combinatorial identities
• Wick's theorem for q-deformed boson operators
• A simple combinatorial interpretation of certain generalized Bell and Stirling numbers
• Dobinski-type relations: Some properties and physical applications
• Combinatorial Solutions to Normal Ordering of Bosons
• $(p,q)$-analogues of the generalized Touchard polynomials and Stirling numbers
• Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers
• Normal ordering problem and the extensions of the Stirling grammar
• Exponential Operators, Dobinski Relations and Summability
• Sheffer Polynomials and the s-ordering of Exponential Boson Operators
• pyBoLaNO: A Python symbolic package for normal ordering involving bosonic ladder operators