## Abstract

The analytical properties of the simple cubic lattice Green function G(t) = $\frac{1}{\pi ^{3}}\mathop{\iiint}_{0}^{\pi}$[t-(cos x$_{1}$ + cos x$_{2}$ + cos x$_{3}$)]$^{-1}$dx$_{1}$dx$_{2}$dx$_{3}$ are investigated. In particular, it is shown that tG(t) can be written in the form tG(t) = [F(9, -$\frac{3}{4}$; $\frac{1}{4}$, $\frac{3}{4}$, 1, $\frac{1}{2}$; 9/t$^{2}$)]$^{2}$, where F(a, b; $\alpha $, $\beta $, $\gamma $, $\delta $; z) denotes a Heun function. The standard analytic continuation formulae for Heun functions are then used to derive various expansions for the Green function G$^{-}$(s) $\equiv $ G$_{\text{R}}$(s) + iG$_{\text{I}}$(s) = $\underset \epsilon \rightarrow 0+\to{\lim}$ G(s - i$\epsilon $) (0 $\leq $ s < $\infty $) about the points s = 0, 1 and 3. From these expansions accurate numerical values of G$_{\text{R}}$(s) and G$_{\text{I}}$(s) are obtained in the range 0 $\leq $ s $\leq $ 3, and certain new summation formulae for Heun functions of unit argument are deduced. Quadratic transformation formulae for the Green function G(t) are discussed, and a connexion between G(t) and the Lame-Wangerin differential equation is established. It is also proved that G(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.