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On the cubic lattice Green functions. (English) Zbl 0808.33015

The author’s abstract: “It is proved that

$K\left({k}_{+}\right)=\left[{\left(4-\eta \right)}^{1/2}-{\left(1-\eta \right)}^{1/2}\right]K\left({k}_{-}\right),$

where $\eta$ is a complex variable which lies in a certain region ${ℛ}_{z}$ 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

${k}_{\mp }^{2}\equiv {k}_{\mp }^{2}\left(\eta \right)=\frac{1}{2}\mp \frac{1}{4}\eta {\left(4-\eta \right)}^{1/2}-\frac{1}{4}\eta {\left(4-\eta \right)}^{1/2}-\frac{1}{4}\left(2-\eta \right){\left(1-\eta \right)}^{1/2}·$

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 $F\left(a,b;\alpha ,\beta ,\gamma ,\delta ;\eta \right)$ 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”.

MSC:
 33E05 Elliptic functions and integrals 33E20 Functions defined by series and integrals 33E30 Functions coming from differential, difference and integral equations 60G50 Sums of independent random variables; random walks