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