Hierarchical Dobiński-type relations via substitution and the moment problem

We consider the transformation properties of integer sequences arising from the normal ordering of exponentiated boson ([a, a†] = 1) monomials of the form exp[λ(a†)ra], r = 1, 2, ..., under the composition of their exponential generating functions. 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 formulae and solve the associated moment problem for several hierarchically defined combinatorial families of sequences.

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