It is proved that
K
(
k
+
) = [(4 –
η
)
1/2
– (1 –
η
)
1/2
]
K
(
k
_), where
η
is a complex variable which lies in a certain region
R
2
of the
η
plane, and
K
(
k
±
) are complete elliptic integrals of the first kind with moduli
k
±
which are given by
k
2
±
Ξ
k
2
±
(
η
) = 1/2 + 1/4
η
(4 –
η
)
1/2
– 1/4 (2 –
η
) (1 –
η
)
1/2
. This basic result is then used to express the face-centred cubic and simple cubic lattice Green functions at the origin in terms of the square of a complete elliptic integral of the first kind. Several new identities involving the Heun function are also derived.
F(a, b; α,β, γ, δ; η)
are also derived. Next it is shown that the three cubic lattice Green functions all have parametric representations which involve the Green function for the two-dimensional honeycomb lattice. Finally, the results are applied to a variety of problems in lattice statistics. In particular, a new simplified formula for the generating function of staircase polygons on a four-dimensional hypercubic lattice is derived.