Penson, K.A.; Blasiak, P.; Duchamp, G.; Horzela, A. and Solomon, A.I.
Hierarchical Dobinski-type relations via substitution and the moment problem.
Journal of Physics A: Mathematical and General, 37(10),
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.
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