ON THE CUBIC LATTICE GREEN-FUNCTIONS

It is proved that K(κ+) = [(4-ν)1/2-(1-ν) 1/2]K(κ), where rj is a complex variable which lies in a certain region R2 of the ν plane, and K(k±) are complete elliptic integrals of the first kind with moduli k± which are given by κ±2 = κ ±2(ν) =1/2 ± 1/4 ν-(4-ν) 1/2-1/4 (2-ν) (1-ν)1/2This 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 F(a, 6;α,β,γ,δ;ν) 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. © 1994 The Royal Society.