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The -Bell numbers
The notion of generalized Bell numbers has appeared in several works but
there is no systematic treatise on this topic. In this paper we fill this gap.
We discuss the most important combinatorial, algebraic and analytic properties
of these numbers which generalize the similar properties of the Bell numbers.
Most of these results seem to be new. It turns out that in a paper of Whitehead
these numbers appeared in a very different context. In addition, we introduce
the so-called -Bell polynomials
Bell polynomials in combinatorial Hopf algebras
Partial multivariate Bell polynomials have been defined by E.T. Bell in 1934.
These polynomials have numerous applications in Combinatorics, Analysis,
Algebra, Probabilities, etc. Many of the formulae on Bell polynomials involve
combinatorial objects (set partitions, set partitions in lists, permutations,
etc.). So it seems natural to investigate analogous formulae in some
combinatorial Hopf algebras with bases indexed by these objects. The algebra of
symmetric functions is the most famous example of a combinatorial Hopf algebra.
In a first time, we show that most of the results on Bell polynomials can be
written in terms of symmetric functions and transformations of alphabets. Then,
we show that these results are clearer when stated in other Hopf algebras (this
means that the combinatorial objects appear explicitly in the formulae). We
investigate also the connexion with the Fa{\`a} di Bruno Hopf algebra and the
Lagrange-B{\"u}rmann formula
Laguerre-type derivatives: Dobinski relations and combinatorial identities
We consider properties of the operators D(r,M)=a^r(a^\dag a)^M (which we call
generalized Laguerre-type derivatives), with r=1,2,..., M=0,1,..., where a and
a^\dag are boson annihilation and creation operators respectively, satisfying
[a,a^\dag]=1. We obtain explicit formulas for the normally ordered form of
arbitrary Taylor-expandable functions of D(r,M) with the help of an operator
relation which generalizes the Dobinski formula. Coherent state expectation
values of certain operator functions of D(r,M) turn out to be generating
functions of combinatorial numbers. In many cases the corresponding
combinatorial structures can be explicitly identified.Comment: 14 pages, 1 figur
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