1,351 research outputs found
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
Hierarchical Dobinski-type relations via substitution and the moment problem
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.Comment: 14 pages, 31 reference
On sums of powers of zeros of polynomials
Due to Girard's (sometimes called Waring's) formula the sum of the th
power of the zeros of every one variable polynomial of degree , ,
can be given explicitly in terms of the coefficients of the monic polynomial. This formula is closely related to a known \par
\noindent variable generalization of Chebyshev's polynomials of the first
kind, . The generating function of these power sums (or moments)
is known to involve the logarithmic derivative of the considered polynomial.
This entails a simple formula for the Stieltjes transform of the distribution
of zeros. Perron-Stieltjes inversion can be used to find this distribution,
{\it e.g.} for .\par Classical orthogonal polynomials are taken as
examples. The results for ordinary Chebyshev and
polynomials are presented in detail. This will correct a statement about power
sums of zeros of Chebyshev's polynomials found in the literature. For the
various cases (Jacobi, Laguerre, Hermite) these moment generating functions
provide solutions to certain Riccati equations
Monomiality principle, Sheffer-type polynomials and the normal ordering problem
We solve the boson normal ordering problem for
with arbitrary functions and and integer , where and
are boson annihilation and creation operators, satisfying
. This consequently provides the solution for the exponential
generalizing the shift operator. In the
course of these considerations we define and explore the monomiality principle
and find its representations. We exploit the properties of Sheffer-type
polynomials which constitute the inherent structure of this problem. In the end
we give some examples illustrating the utility of the method and point out the
relation to combinatorial structures.Comment: Presented at the 8'th International School of Theoretical Physics
"Symmetry and Structural Properties of Condensed Matter " (SSPCM 2005),
Myczkowce, Poland. 13 pages, 31 reference
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