Application of Mahler measure theory to the face-centred cubic lattice Green function at the origin and its associated logarithmic integral

摘要

The mathematical properties of the face-centred cubic lattice Green function G(w) ≡ 1/π ~3 ∫ ~π _0 ∫ π _0 ∫ π _0 dθ _1 dθ _2 dθ _3/w - c(θ1) c(θ _2) - c(θ _2) c(θ _3) - c(θ _3) c(θ _1) and the associated logarithmic integral S(w) ≡ 1/π ~3 ∫ ~π _0 ∫ π _0 ∫ π _0 ln[w - c(θ _1) c(θ _2) - c(θ _2) c(θ _3) -c(θ _3) c(θ _1)] dθ _1 dθ _2 dθ _3 are investigated, where c(θ) ≡ cos(θ) and w = w 1+iw 2 is a complex variable in the w plane. In particular, the theory of Mahler measures is used to obtain a closed-form expression for S(w) in terms of 5F_4 generalized hypergeometric functions. The method of analytic continuation is then applied to this result in order to prove that S(3) = -14/15 ln 2 + 8/5 ln 3 8/135 5F 4 [4/3, 3/2, 5/3, 1, 1; 1 2, 2, 2, 2]. Next the relation dS/dw = G(w), where w = (3,+∞), is used to derive the simple formula G(w) = 1/w { _2F ~1 1/6, 1/3; 1; 27(w + 1)/4w ~3}2, where w lies in a restricted region R1 of the cut w plane. The limit function G- (w1) ≡ lim ε → 0+ G(w1 iε) = GR(w1) + iGI(w1) is also evaluated in the intervals w1 ε (-1, 0] and w1 ε (0, 3]. It is shown that GR(w1) and GI(w1) can be expressed in terms of 2F1[z(w1)] hypergeometric functions, where the independent variable z(w1) is a real-valued rational function of w 1. Finally, new formulae are derived for the number of random walks r _n on the face-centred cubic lattice which return to their starting point (not necessarily for the first time) after n nearest-neighbour steps.