*(English)*Zbl 0808.33015

The author’s abstract: “It is proved that

where $\eta $ is a complex variable which lies in a certain region ${\mathcal{R}}_{z}$ of the $\eta $ plain, and $K\left({k}_{\mp}\right)$ are complete elliptic integrals of the first kind with moduli ${k}_{\mp}$ which are given by

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”.