  We study the partially asymmetric exclusion process with open boundaries. We
generalise the matrix approach previously used to solve the special case of
total asymmetry and derive exact expressions for the partition sum and currents
valid for all values of the asymmetry parameter q. Due to the relationship
between the matrix algebra and the q-deformed quantum harmonic oscillator
algebra we find that q-Hermite polynomials, along with their orthogonality
properties and generating functions, are of great utility. We employ two
distinct sets of q-Hermite polynomials, one for q<1 and the other for q>1. It
turns out that these correspond to two distinct regimes: the previously studied
case of forward bias (q<1) and the regime of reverse bias (q>1) where the
boundaries support a current opposite in direction to the bulk bias. For the
forward bias case we confirm the previously proposed phase diagram whereas the
case of reverse bias produces a new phase in which the current decreases
exponentially with system size.
