We introduce a spectral transform for the finite relativistic Toda lattice
(RTL) in generalized form. In the nonrelativistic case, Moser constructed a
spectral transform from the spectral theory of symmetric Jacobi matrices. Here
we use a non-symmetric generalized eigenvalue problem for a pair of bidiagonal
matrices (L,M) to define the spectral transform for the RTL. The inverse
spectral transform is described in terms of a terminating T-fraction. The
generalized eigenvalues are constants of motion and the auxiliary spectral data
have explicit time evolution. Using the connection with the theory of Laurent
orthogonal polynomials, we study the long-time behaviour of the RTL. As in the
case of the Toda lattice the matrix entries have asymptotic limits. We show
that L tends to an upper Hessenberg matrix with the generalized eigenvalues
sorted on the diagonal, while M tends to the identity matrix.Comment: 24 pages, 9 figure