On the simple cubic lattice Green function

Abstract

The analytical properties of the simple cubic lattice Green functionG(t)=1π3∫∫∫0π[t−(cosx1+cosx2+cosx3)]−1dx1dx2dx3are investigated. In particular, it is shown that tG(t) can be written in the formtG(t)=[F(9,−34;14,34,1,12;9/t2)]2,whereF(a,b;α, β, γ, β; z) denotes a Heun function. The standard analytic continuation formulae for Heun functions are then used to derive various expansions for the Green functionG−(s)≡GR(s)+iGI(s)=lim∈→0+G(s−i∈)(0≤s<∞)about the pointss= 0,1 and 3. From these expansions accurate numerical values ofG_R(s) andG_I(s) are obtained in the range 0≤s≤3, and certain new summation formulae for Heun functions of unit argum ent are deduced. Quadratic transformation formulae for the Green functionG(t)are discussed, and a connexion betweenG(t)and the Lamé-Wangerin differential equation is established. It is also proved thatG(t)can be expressed as a product of two complete elliptic integrals of the first kind. Finally, several applications of the results are made in lattice statistics.