Exact closed form of the return probability on the Bethe lattice

An exact closed-form solution for the return probability of a random walk on the Bethe lattice is given. The long-time asymptotic form confirms a previously known expression. It is, however, shown that this exact result reduces to the proper expression when the Bethe lattice degenerates on a line, unlike the asymptotic result which is singular. This is shown to be an artefact of the asymptotic expansion. The density of states is also calculated.