  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.
