25 research outputs found
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
Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle
With the aim of derive a quasi-monomiality formulation in the context of
discrete hypercomplex variables, one will amalgamate through a
Clifford-algebraic structure of signature the umbral calculus framework
with Lie-algebraic symmetries. The exponential generating function ({\bf EGF})
carrying the {\it continuum} Dirac operator D=\sum_{j=1}^n\e_j\partial_{x_j}
together with the Lie-algebraic representation of raising and lowering
operators acting on the lattice h\BZ^n is used to derive the corresponding
hypercomplex polynomials of discrete variable as Appell sets with membership on
the space Clifford-vector-valued polynomials. Some particular examples
concerning this construction such as the hypercomplex versions of falling
factorials and the Poisson-Charlier polynomials are introduced. Certain
applications from the view of interpolation theory and integral transforms are
also discussed.Comment: 24 pages. 1 figure. v2: a major revision, including numerous
improvements throughout the paper was don
Bernoulli type polynomials on Umbral Algebra
The aim of this paper is to investigate generating functions for modification
of the Milne-Thomson's polynomials, which are related to the Bernoulli
polynomials and the Hermite polynomials. By applying the Umbral algebra to
these generating functions, we provide to deriving identities for these
polynomials