On the cubic lattice Green functions

Wheatstone Physics Laboratory, King's College, University of London, Strand, London WC2R 2LS, U.K. It is proved that K (k$_{+}$) = [(4-$\eta $)$^{\frac{1}{2}}$ - (1 - $\eta $)$^{\frac{1}{2}}$]K(k$_{-}$), where $\eta $ is a complex variable which lies in a certain region $\scr{R}_{2}$ of the $\eta $ plane, and K (k$_{\pm}$) are complete elliptic integrals of the first kind with moduli k$_{\pm}$ which are given by k$_{\pm}^{2}\equiv $ k$_{\pm}^{2}$($\eta $) = $\frac{1}{2}$ $\pm \frac{1}{4}\eta $(4 - $\eta $)$^{\frac{1}{2}}$ - $\frac{1}{4}$(2-$\eta $)(1-$\eta $)$^{\frac{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 F(a, b; $\alpha $, $\beta $, $\gamma $, $\delta $; $\eta $) 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.