  We obtain analytic solutions for the one-dimensional Dirac equation with the
Morse potential as an infinite series of square integrable functions. These
solutions are for all energies, the discrete as well as the continuous. The
elements of the spinor basis are written in terms of the confluent
hypergeometric functions. They are chosen such that the matrix representation
of the Dirac-Morse operator for continuous spectrum (i.e., for scattering
energies larger than the rest mass) is tridiagonal. Consequently, the wave
equation results in a three-term recursion relation for the expansion
coefficients of the wavefunction. The solution of this recursion relation is
obtained in terms of the continuous dual Hahn orthogonal polynomials. On the
other hand, for the discrete spectrum (i.e., for bound states with energies
less than the rest mass) the spinor wave functions result in a diagonal matrix
representation for the Dirac-Morse Hamiltonian.
