Dobinski-type relations and the Log-normal distribution

Abstract

We consider sequences of generalized Bell numbers B(n), n=0,1,... for which there exist Dobinski-type summation formulas; that is, where B(n) is represented as an infinite sum over k of terms P(k)^n/D(k). These include the standard Bell numbers and their generalizations appearing in the normal ordering of powers of boson monomials, as well as variants of the "ordered" Bell numbers. For any such B we demonstrate that every positive integral power of B(m(n)), where m(n) is a quadratic function of n with positive integral coefficients, is the n-th moment of a positive function on the positive real axis, given by a weighted infinite sum of log-normal distributions.

7 pages, 2 Figures

References in corpus (4)

• New q-deformed coherent states with an explicitly known resolution of unity
• Generalized Coherent States Associated with the $C_{\lambda }$-Extended Oscillator
• Generalized coherent states associated with the C_λ-extended oscillator
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Cited by in corpus (5)

• Combinatorics of boson normal ordering and some applications
• Hierarchical Dobinski-type relations via substitution and the moment problem
• Laguerre-type derivatives: Dobinski relations and combinatorial identities
• Combinatorial coherent states via normal ordering of bosons
• Dobinski-type relations: Some properties and physical applications