  The analysis of the return probability is one of the most essential and
fundamental topics in the study of classical random walks. In this paper, we
study the return probability of quantum and correlated random walks in the
one-dimensional integer lattice by the path counting method. We show that the
return probability of both quantum and correlated random walks can be expressed
in terms of the Legendre polynomial. Moreover, the generating function of the
return probability can be written in terms of elliptic integrals of the first
and second kinds for the quantum walk.
