41 research outputs found
Laguerre functions and representations of su(1,1)
Spectral analysis of a certain doubly infinite Jacobi operator leads to
orthogonality relations for confluent hypergeometric functions, which are
called Laguerre functions. This doubly infinite Jacobi operator corresponds to
the action of a parabolic element of the Lie algebra . The
Clebsch-Gordan coefficients for the tensor product representation of a positive
and a negative discrete series representation of are
determined for the parabolic bases. They turn out to be multiples of Jacobi
functions. From the interpretation of Laguerre polynomials and functions as
overlap coefficients, we obtain a product formula for the Laguerre polynomials,
given by a discontinuous integral over Laguerre functions, Jacobi functions and
continuous dual Hahn polynomials.Comment: 19 page
Quantum Analogs of Tensor Product Representations of su(1,1)
We study representations of that can be considered as quantum
analogs of tensor products of irreducible *-representations of the Lie algebra
. We determine the decomposition of these representations into
irreducible *-representations of by diagonalizing the action of
the Casimir operator on suitable subspaces of the representation spaces. This
leads to an interpretation of the big -Jacobi polynomials and big -Jacobi
functions as quantum analogs of Clebsch-Gordan coefficients