  A celebrated result in probability theory is that a simple symmetric random
walk on the $d$-dimensional lattice $\mathbb{Z}^d$ is recurrent for $d=1,2$ and
transient for $d\geq 3$. In this note, we derive a closed-form expression, in
terms of the Lauricella function $F_C$, for the return probability for all
$d\geq3$. Previously, a closed-form formula had only been available for $d=3$.
