  We demonstrate an alternative method for calculating the asymptotic behaviour
of the discrete one-coin quantum walk on the infinite line, via the Jacobi
polynomials that arise in the path integral representation. This is
significantly easier to use than the Darboux method. It also provides a single
integral representation for the wavefunction that works over the full range of
positions, $n,$ including throughout the transitional range where the behaviour
changes from oscillatory to exponential. Previous analyses of this system have
run into difficulties in the transitional range, because the approximations on
which they were based break down here. The fact that there are two different
kinds of approach to this problem (Path Integral vs. Schr\"{o}dinger wave
mechanics) is ultimately a manifestation of the equivalence between the
path-integral formulation of quantum mechanics and the original formulation
developed in the 1920s. We discuss how and why our approach is related to the
two methods that have already been used to analyse these systems.
