On Pólya's random walk constants

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 $d\ge 3$. In this note, we derive a closed-form expression, in terms of the Lauricella function $F_C$, for the return probability for all $d\ge 3$. Previously, a closed-form formula had only been available for $d=3$.