Dobiński-type relations: Some properties and physical applications

We introduce a generalization of the Dobiński relation through which we define a family of Bell-type numbers and polynomials. For all these sequences we find the weight function of the moment problem and give their generating functions. We provide a physical motivation of this extension in the context of the boson normal ordering problem and its relation to an extension of the Kerr Hamiltonian. Dobiński-type relations: Some properties and physical applications 2 In this note we consider the Dobiński relation and its generalization. This topic naturally belongs to the field of combinatorial analysis. The Dobiński relation [1] was first derived in connection with Bell numbers B(n) = 1, 1, 2, 5, 52, 203, 877, . . ., n = 0, 1, 2, ..., which describe partitions of a set [2],[3]. That remarkable formula represents the integer sequence B(n) as an infinite sum of ratios B(n) = e−1 ∞