583 research outputs found
The Jacobi inversion formula
We look for spectral type differential equations satisfied by the generalized
Jacobi polynomials which are orthogonal on the interval [-1,1] with respect to
a weight function consisting of the classical Jacobi weight function together
with point masses at the endpoints of the interval of orthogonality.
In order to find explicit formulas for the coefficients of these differential
equations we have to solve systems of equations involving derivatives of the
classical Jacobi polynomials. These systems of equations have a unique solution
which can be given explicitly in terms of Jacobi polynomials. This is a
consequence of the Jacobi inversion formula which is proved in this paper.Comment: 15 page
On differential equations for Sobolev-type Laguerre polynomials
We obtain all spectral type differential equations satisfied by the
Sobolev-type Laguerre polynomials. This generalizes the results found in 1990
by the first and second author in the case of the generalized Laguerre
polynomials defined by T.H. Koornwinder in 1984.Comment: 45 page
Differential equations for generalized Jacobi polynomials
We look for spectral type differential equations satisfied by the generalized
Jacobi polynomials, which are orthogonal on the interval [-1,1] with respect to
a weight function consisting of the classical Jacobi weight function together
with two point masses at the endpoints of the interval of orthogonality.
We show that such a differential equation is uniquely determined and we give
explicit representations for the coefficients.
In case of nonzero mass points the order of this differential equation is
infinite, except for nonnegative integer values of (one of) the parameters.
Otherwise, the finite order is explictly given in terms of the parameters.Comment: 33 pages, submitted for publicatio
Wilson Polynomials and the Lorentz Transformation Properties of the Parity Operator
The parity operator for a parity-symmetric quantum field theory transforms as
an infinite sum of irreducible representations of the homogeneous Lorentz
group. These representations are connected with Wilson polynomials
The algebra of dual -1 Hahn polynomials and the Clebsch-Gordan problem of sl_{-1}(2)
The algebra H of the dual -1 Hahn polynomials is derived and shown to arise
in the Clebsch-Gordan problem of sl_{-1}(2). The dual -1 Hahn polynomials are
the bispectral polynomials of a discrete argument obtained from a q-> -1 limit
of the dual q-Hahn polynomials. The Hopf algebra sl_{-1}(2) has four generators
including an involution, it is also a q-> -1 limit of the quantum algebra
sl_{q}(2) and furthermore, the dynamical algebra of the parabose oscillator.
The algebra H, a two-parameter generalization of u(2) with an involution as
additional generator, is first derived from the recurrence relation of the -1
Hahn polynomials. It is then shown that H can be realized in terms of the
generators of two added sl_{-1}(2) algebras, so that the Clebsch-Gordan
coefficients of sl_{-1}(2) are dual -1 Hahn polynomials. An irreducible
representation of H involving five-diagonal matrices and connected to the
difference equation of the dual -1 Hahn polynomials is constructed.Comment: 15 pages, Some minor changes from version #
On the Krall-type Askey-Wilson Polynomials
In this paper the general Krall-type Askey-Wilson polynomials are introduced.
These polynomials are obtained from the Askey-Wilson polynomials via the
addition of two mass points to the weight function of them at the points
. Several properties of such new family are considered, in particular the
three-term recurrence relation and the representation as basic hypergeometric
series
New connection formulae for some q-orthogonal polynomials in q-Askey scheme
New nonlinear connection formulae of the q-orthogonal polynomials, such
continuous q-Laguerre, continuous big q-Hermite, q-Meixner-Pollaczek and
q-Gegenbauer polynomials, in terms of their respective classical analogues are
obtained using a special realization of the q-exponential function as infinite
multiplicative series of ordinary exponential function
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