Laguerre functions and representations of su(1,1)

Details Laguerre functions and representations of su(1,1) Laguerre functions and representations of su(1,1)

2003-02-27

arxiv.org/abs/math/0302342v1

Indag. Math. (N.S.) 14 (2003), no. 3-4, 329-352

Mathematics

Classical Analysis and ODEs

19 pages

Scientific paper

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 $\mathfrak{su}(1,1)$. The Clebsch-Gordan coefficients for the tensor product representation of a positive and a negative discrete series representation of $\mathfrak{su}(1,1)$ 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.

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