Exact product forms for the simple cubic lattice Green function II

The analytical properties of the lattice Green function G(2n, n, n; w) 1/π3 ∫0π∫0π∫0π (cos 2nθ1 cos nθ2 cos nθ3)/(w - cos θ1 - cos θ2 - cos θ3) dθ1 dθ2 dθ3 are investigated, where n is an integer and w is a complex variable. In particular, it is shown that G(2n, n, n; w) is a solution of a fourth-order linear differential equation of the Fuchsian type. From this differential equation it is found that G(2n, n, n; w) can be evaluated in terms of a product of two Heun functions {Hj(n, v): j = 1, 2}, where v ≡ v(w) = 1/w2 (1 + √(1-1/w2))-1 (1 + √(1-9/w2))-1. A detailed discussion of the properties of {Hj(n, v): j = 1, 2} is then given. The Heun function results are used to prove that the product form for G(2n, n, n; w) can be expressed in terms of complete elliptic integrals of the first and second kinds. It is also demonstrated that G(2n, n, n; w) can be written in the hypergeometric form wG(2n, n, n; w) ((1/4)n(3/4)n)/(n!)2 (w2/(w2+3))1/2 [w2/8 (√(1-1/w2) - √(1-9/w2))2]2n x 2F1 (1/4, 3/4; n+1; η+) 2F1 (1/4, 3/4; n+1; η-) where η± ≡ η±(w) 1/2 + w2/(2(3+w2)2) √(1 - 1/w2) [±16 + (5-w2)√(1 - 9/w2)] and (a)n denotes the Pochhammer symbol. This formula is valid for varying values of w in the neighbourhood of w = ∞, provided that the argument function η+(w) does not take real values in the interval (1, + ∞). Finally, this 2F1 product form is used to determine the asymptotic behaviour of G(2n, n, n; w) as n → ∞