686 research outputs found
Extensions of discrete classical orthogonal polynomials beyond the orthogonality
It is well known that the family of Hahn polynomials
is orthogonal with respect to a certain
weight function up to . In this paper we present a factorization for Hahn
polynomials for a degree higher than and we prove that these polynomials
can be characterized by a -Sobolev orthogonality.
We also present an analogous result for dual-Hahn, Krawtchouk, and Racah
polynomials and give the limit relations between them for all n\in \XX N_0.
Furthermore, in order to get this results for the Krawtchouk polynomials we
will get a more general property of orthogonality for Meixner polynomials.Comment: 2 figures, 20 page
Symmetric coupling of angular momenta, quadratic algebras and discrete polynomials
Eigenvalues and eigenfunctions of the volume operator, associated with the
symmetric coupling of three SU(2) angular momentum operators, can be analyzed
on the basis of a discrete Schroedinger-like equation which provides a
semiclassical Hamiltonian picture of the evolution of a `quantum of space', as
shown by the authors in a recent paper. Emphasis is given here to the
formalization in terms of a quadratic symmetry algebra and its automorphism
group. This view is related to the Askey scheme, the hierarchical structure
which includes all hypergeometric polynomials of one (discrete or continuous)
variable. Key tool for this comparative analysis is the duality operation
defined on the generators of the quadratic algebra and suitably extended to the
various families of overlap functions (generalized recoupling coefficients).
These families, recognized as lying at the top level of the Askey scheme, are
classified and a few limiting cases are addressed.Comment: 10 pages, talk given at "Physics and Mathematics of Nonlinear
Phenomena" (PMNP2013), to appear in J. Phys. Conf. Serie
Wilson function transforms related to Racah coefficients
The irreducible -representations of the Lie algebra consist of
discrete series representations, principal unitary series and complementary
series. We calculate Racah coefficients for tensor product representations that
consist of at least two discrete series representations. We use the explicit
expressions for the Clebsch-Gordan coefficients as hypergeometric functions to
find explicit expressions for the Racah coefficients. The Racah coefficients
are Wilson polynomials and Wilson functions. This leads to natural
interpretations of the Wilson function transforms. As an application several
sum and integral identities are obtained involving Wilson polynomials and
Wilson functions. We also compute Racah coefficients for U_q(\su(1,1)), which
turn out to be Askey-Wilson functions and Askey-Wilson polynomials.Comment: 48 page
Cyclic tridiagonal pairs, higher order Onsager algebras and orthogonal polynomials
The concept of cyclic tridiagonal pairs is introduced, and explicit examples
are given. For a fairly general class of cyclic tridiagonal pairs with
cyclicity N, we associate a pair of `divided polynomials'. The properties of
this pair generalize the ones of tridiagonal pairs of Racah type. The algebra
generated by the pair of divided polynomials is identified as a higher-order
generalization of the Onsager algebra. It can be viewed as a subalgebra of the
q-Onsager algebra for a proper specialization at q the primitive 2Nth root of
unity. Orthogonal polynomials beyond the Leonard duality are revisited in light
of this framework. In particular, certain second-order Dunkl shift operators
provide a realization of the divided polynomials at N=2 or q=i.Comment: 32 pages; v2: Appendices improved and extended, e.g. a proof of
irreducibility is added; v3: version for Linear Algebra and its Applications,
one assumption added in Appendix about eq. (A.2
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