Generating Functions for Asymmetric Random Walk Processes With Double Absorbing Barriers

Generating functions for asymmetric step-size paths restricted by two absorbing barriers are derived. The method begins by applying the Lagrange inversion formula to arbitrary powers of roots of the characteristic equation, that being a trinomial, which produces generating function as function (z) of the conditional probability of absorption of a particle, on a path restricted by two absorbing barriers. The exact enumeration of an asymmetric walk with two absorbing barriers is given.