Wolter Groenevelt

The Wilson function transform

Two unitary integral transforms with a very-well poised

MEIXNER FUNCTIONS AND POLYNOMIALS RELATED TO LIE ALGEBRA REPRESENTATIONS

_7F_6

Continuous Hahn functions as Clebsch-Gordan coefficients

-function as a kernel are given. For both integral transforms the inverse is the same as the original transform after an involution on the parameters. The

Quantum Analogs of Tensor Product Representations of su(1;1) ?

_7F_6

The indeterminate moment problem for the q-Meixner polynomials

-function involved can be considered as a non-polynomial extension of the Wilson polynomial, and is therefore called a Wilson function. The two integral transforms are called a Wilson function transform of type I and type II. Furthermore, a few explicit transformations of hypergeometric functions are calculated, and it is shown that the Wilson function transform of type I maps a basis of orthogonal polynomials onto a similar basis of polynomials.

The Dual Quantum Group for the Quantum Group Analog of the Normalizer of SU(1, 1) in

The decomposition of the tensor product of a positive and a negative discrete series representation of the Lie algebra (1,1) is a direct integral over the principal unitary series representations. In the decomposition discrete terms can occur, and these are a finite number of discrete series representations or one complementary series representation. The interpretation of Meixner functions and polynomials as overlap coefficients in the four classes of representations and the Clebsch–Gordan decomposition, lead to a general bilinear generating function for the Meixner polynomials. Finally, realizing the positive and negative discrete series representations as operators on the spaces of holomorphic and anti-holomorphic functions, respectively, a non-symmetric type Poisson kernel is found for the Meixner functions.

Laguerre functions and representations of su(1,1)

An explicit bilinear generating function for Meixner-Pollaczek polynomials is proved. This formula involves continuous dual Hahn polynomials, Meixner-Pollaczek functions, and non-polynomial 3 F 2-hypergeometric functions that we consider as continuous Hahn functions. An integral transform pair with continuous Hahn functions as kernels is also proved. These results have an interpretation for the tensor product decomposition of a positive and a negative discrete series representation of su(1, 1) with respect to hyperbolic bases, where the Clebsch-Gordan coefficients are continuous Hahn functions.

A hypergeometric function transform and matrix-valued orthogonal polynomials

Abstract. We study representations of Uq(su(1; 1)) that can be considered as quantum analogs of tensor products of irreducible -representations of the Lie algebra su(1; 1). We determine the decomposition of these representations into irreducible -representations of Uq(su(1; 1)) by diagonalizing the action of the Casimir operator on suitable subspaces of the representation spaces. This leads to an interpretation of the big q-Jacobi polynomials and big q-Jacobi functions as quantum analogs of Clebsch { Gordan coefficients.

Orthogonal Stochastic Duality Functions from Lie Algebra Representations

For a class of orthogonal polynomials related to the q-Meixner polynomials corresponding to an indeterminate moment problem we give a one-parameter family of orthogonality measures. For these measures we complement the orthogonal polynomials to an orthogonal basis for the corresponding weighted L^2-space explicitly. The result is proved in two ways; by a spectral decomposition of a suitable operator and by direct series manipulation. We discuss extensions to explicit non-positive measures and the relation to other indeterminate moment problems for the continuous dual q^-^1-Hahn and q-Laguerre polynomials.

Coupling coefficients for tensor product representations of quantum SU(2)

The quantum group analog of the normalizer of SU(1, 1) in is an important and nontrivial example of a noncompact quantum group. The general theory of locally compact quantum groups in the operator algebra setting implies the existence of the dual quantum group. The first main goal of this article is to give an explicit description of the dual quantum group for this example involving the quantized enveloping algebra . It turns out that does not suffice to generate the dual quantum group. The dual quantum group is graded with respect to commutation and anticommutation with a suitable analog of the Casimir operator characterized by an affiliation relation to a von Neumann algebra. This is used to obtain an explicit set of generators. Having the dual quantum group the left regular corepresentation of the quantum group analog of the normalizer of SU(1, 1) in is decomposed into irreducible corepresentations. Upon restricting the irreducible corepresentations to -representation one finds combinations of the positive and negative discrete series representations with the strange series representations as well as combinations of the principal unitary series representations. The detailed analysis of this example involves the analysis of special functions of basic hypergeometric type and, in particular, some results on these special functions are obtained, which are stated separately. This article is split into two parts: the first part gives almost all of the statements and the results, and the statements of this part are independent of the second part. The second part contains the proofs of all the statements.