Hierarchical Dobi ski-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 †) r a ], 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 (Dobiński-type relations). We present a number of examples of such composition satisfying properties ( a ) and ( b ). We obtain new Dobiński-type formulae and solve the associated moment problem for several hierarchically defined combinatorial families of sequences.