Hierarchical Dobinski-type relations via substitution and the moment problem

Abstract

We consider the transformation properties of integer sequences arising from the normal ordering of exponentiated boson ([a,a*]=1) monomials of the form exp(x (a*)^r a), r=1,2,..., under the composition of their exponential generating functions (egf). They turn out to be of Sheffer-type. We demonstrate that two key properties of these sequences remain preserved under substitutional composition: (a)the property of being the solution of the Stieltjes moment problem; and (b) the representation of these sequences through infinite series (Dobinski-type relations). We present a number of examples of such composition satisfying properties (a) and (b). We obtain new Dobinski-type formulas and solve the associated moment problem for several hierarchically defined combinatorial families of sequences.

14 pages, 31 references

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• The general boson normal ordering problem
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• Hankel hyperdeterminants and Selberg integrals
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Cited by in corpus (8)

• Boson Normal Ordering via Substitutions and Sheffer-type Polynomials
• Combinatorics of boson normal ordering and some applications
• Calculation of some determinants using the s-shifted factorial
• Symbolic methods for the evaluation of sum rules of Bessel functions
• Dobinski-type relations: Some properties and physical applications
• From Quantum Mechanics to Quantum Field Theory: The Hopf route
• Free quasi-symmetric functions, product actions and quantum field theory of partitions
• On powers of Stirling matrices