Exact product form for the anisotropic simple cubic lattice Green function

Abstract

The analytical properties of the lattice Green function are investigated, where n is an integer, w is a complex variable and α is a real parameter in the interval (0, ∞). In particular, it is shown that G(2n, n, n; α, w) is a solution of a fourth-order linear differential equation of the Fuchsian type. From this differential equation, it is proved that G(2n, n, n; α, w) can be expressed in the hypergeometric form where and (β)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+(α, w) does not take real values in the interval (1, ∞). The 2F1 product form is used to determine the asymptotic behaviour of G(2n, n, n; α, w) as n → ∞. Finally, a five-term linear recurrence relation is given for G(2n, n, n; α, w).