  In random walks, the path representation of the Green's function is an
infinite sum over the length of path probability density functions (PDFs). Here
we derive and solve, in Laplace space, the recursion relation for the n order
path PDF for any arbitrarily inhomogeneous semi-Markovian random walk in a
one-dimensional (1D) chain of L states. The recursion relation relates the n
order path PDF to L/2 (round towards zero for an odd L) shorter path PDFs, and
has n independent coefficients that obey a universal formula. The z transform
of the recursion relation straightforwardly gives the generating function for
path PDFs, from which we obtain the Green's function of the random walk, and
derive an explicit expression for any path PDF of the random walk. These
expressions give the most detailed description of arbitrarily inhomogeneous
semi-Markovian random walks in 1D.
