1,429 research outputs found
Doubling (Dual) Hahn Polynomials: Classification and Applications
We classify all pairs of recurrence relations in which two Hahn or dual Hahn
polynomials with different parameters appear. Such couples are referred to as
(dual) Hahn doubles. The idea and interest comes from an example appearing in a
finite oscillator model [Jafarov E.I., Stoilova N.I., Van der Jeugt J., J.
Phys. A: Math. Theor. 44 (2011), 265203, 15 pages, arXiv:1101.5310]. Our
classification shows there exist three dual Hahn doubles and four Hahn doubles.
The same technique is then applied to Racah polynomials, yielding also four
doubles. Each dual Hahn (Hahn, Racah) double gives rise to an explicit new set
of symmetric orthogonal polynomials related to the Christoffel and Geronimus
transformations. For each case, we also have an interesting class of
two-diagonal matrices with closed form expressions for the eigenvalues. This
extends the class of Sylvester-Kac matrices by remarkable new test matrices. We
examine also the algebraic relations underlying the dual Hahn doubles, and
discuss their usefulness for the construction of new finite oscillator models
Induced Representations of the Two Parametric Quantum Deformation U[gl(2/2)]
The two-parametric quantum superalgebra and its induced
representations are considered. A method for constructing all
finite-dimensional irreducible representations of this quantum superalgebra is
also described in detail. It turns out that finite-dimensional representations
of the two-parametric , even at generic deformation
parameters, are not simply trivial deformations from those of the classical
superalgebra , unlike the one-parametric cases.Comment: Latex, 40 pages, no figure. To appear in J. Math. Phys. 41 (2000
Dynamical supersymmetry of spin particle-magnetic field interaction
We study the super and dynamical symmetries of a fermion in a monopole
background. The Hamiltonian also involves an additional spin-orbit coupling
term, which is parameterized by the gyromagnetic ratio. We construct the
superinvariants associated with the system using a SUSY extension of a
previously proposed algorithm, based on Grassmann-valued Killing tensors.
Conserved quantities arise for certain definite values of the gyromagnetic
factor: SUSY requires ; a Kepler-type dynamical symmetry only
arises, however, for the anomalous values and . The two anomalous
systems can be unified into an SUSY system built by doubling the number
of Grassmann variables. The planar system also exhibits an supersymmetry
without Grassmann variable doubling.Comment: 23 page
More on the q-oscillator algebra and q-orthogonal polynomials
Properties of certain -orthogonal polynomials are connected to the
-oscillator algebra. The Wall and -Laguerre polynomials are shown to
arise as matrix elements of -exponentials of the generators in a
representation of this algebra. A realization is presented where the continuous
-Hermite polynomials form a basis of the representation space. Various
identities are interpreted within this model. In particular, the connection
formula between the continuous big -Hermite polynomials and the continuous
-Hermite polynomials is thus obtained, and two generating functions for
these last polynomials are algebraically derived
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