Slow variation and uniqueness of solutions to the functional equation in the branching random walk

In this short communication, some of the recent results of Liu (1998) and Biggins and Kyprianou (1997), concerning solutions to a certain functional equation associated with the branching random walk, are strengthened. Their importance is emphasized in the context of travelling wave solutions to a discrete version of the KPP equation and the connection with the behaviour of the rightmost particle in the n th generation.