Normally ordered forms of powers of differential operators and their combinatorics

We investigate the combinatorics of the general formulas for the powers of the operator h∂k, where h is a central element of a ring and ∂ is a differential operator. This generalizes previous work on the powers of operators h∂. New formulas for the generalized Stirling numbers are obtained.Ministerio de Economía y competitividad MTM2016-75024-PJunta de Andalucía P12-FQM-2696Junta de Andalucía FQM–33

An algebraic approach to Polya processes

P\'olya processes are natural generalization of P\'olya-Eggenberger urn models. This article presents a new approach of their asymptotic behaviour {\it via} moments, based on the spectral decomposition of a suitable finite difference operator on polynomial functions. Especially, it provides new results for {\it large} processes (a P\'olya process is called {\it small} when 1 is simple eigenvalue of its replacement matrix and when any other eigenvalue has a real part $\leq 1/2$; otherwise, it is called large)

On the Initial-Value Problem to the Quantum Dual BBGKY Hierarchy

We develop a rigorous formalism for the description of the evolution of observables in quantum systems of particles. We construct a solution of the initial-value problem to the quantum dual BBGKY hierarchy of equations as an expansion over particle clusters whose evolution are governed by the corresponding-order dual cumulant (dual semi-invariant) of the evolution operators of finitely many particles. For initial data from the space of sequences of bounded operators the existence and uniqueness theorem is proved.Comment: 22 page