43 research outputs found
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
Quantum function algebras as quantum enveloping algebras
Inspired by a result in [Ga], we locate two -integer forms of
, along with a presentation by generators and relations, and
prove that for they specialize to , where is the Lie bialgebra of the Poisson Lie group dual of ; moreover, we explain the relation with [loc. cit.]. In sight of
this, we prove two PBW-like theorems for , both related to the
classical PBW theorem for .Comment: 27 pages, AMS-TeX C, Version 3.0 - Author's file of the final
version, as it appears in the journal printed version, BUT for a formula in
Subsec. 3.5 and one in Subsec. 5.2 - six lines after (5.1) - that in this
very pre(post)print have been correcte
LU factorizations, q=0 limits, and p-adic interpretations of some q-hypergeometric orthogonal polynomials
For little q-Jacobi polynomials and q-Hahn polynomials we give particular
q-hypergeometric series representations in which the termwise q=0 limit can be
taken. When rewritten in matrix form, these series representations can be
viewed as LU factorizations. We develop a general theory of LU factorizations
related to complete systems of orthogonal polynomials with discrete
orthogonality relations which admit a dual system of orthogonal polynomials.
For the q=0 orthogonal limit functions we discuss interpretations on p-adic
spaces. In the little 0-Jacobi case we also discuss product formulas.Comment: changed title, references updated, minor changes matching the version
to appear in Ramanujan J.; 22 p
On a correspondence between quantum SU(2), quantum E(2) and extended quantum SU(1,1)
In a previous paper, we showed how one can obtain from the action of a
locally compact quantum group on a type I-factor a possibly new locally compact
quantum group. In another paper, we applied this construction method to the
action of quantum SU(2) on the standard Podles sphere to obtain Woronowicz'
quantum E(2). In this paper, we will apply this technique to the action of
quantum SU(2) on the quantum projective plane (whose associated von Neumann
algebra is indeed a type I-factor). The locally compact quantum group which
then comes out at the other side turns out to be the extended SU(1,1) quantum
group, as constructed by Koelink and Kustermans. We also show that there exists
a (non-trivial) quantum groupoid which has at its corners (the duals of) the
three quantum groups mentioned above.Comment: 35 page
Free q-Schrodinger Equation from Homogeneous Spaces of the 2-dim Euclidean Quantum Group
After a preliminary review of the definition and the general properties of
the homogeneous spaces of quantum groups, the quantum hyperboloid qH and the
quantum plane qP are determined as homogeneous spaces of Fq(E(2)). The
canonical action of Eq(2) is used to define a natural q-analog of the free
Schro"dinger equation, that is studied in the momentum and angular momentum
bases. In the first case the eigenfunctions are factorized in terms of products
of two q-exponentials. In the second case we determine the eigenstates of the
unitary representation, which, in the qP case, are given in terms of Hahn-Exton
functions. Introducing the universal T-matrix for Eq(2) we prove that the
Hahn-Exton as well as Jackson q-Bessel functions are also obtained as matrix
elements of T, thus giving the correct extension to quantum groups of well
known methods in harmonic analysis.Comment: 19 pages, plain tex, revised version with added materia
Fermat-linked relations for the Boubaker polynomial sequences via Riordan matrices analysis
The Boubaker polynomials are investigated in this paper. Using Riordan
matrices analysis, a sequence of relations outlining the relations with
Chebyshev and Fermat polynomials have been obtained. The obtained expressions
are a meaningful supply to recent applied physics studies using the Boubaker
polynomials expansion scheme (BPES).Comment: 12 pages, LaTe