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

作者：Joyce G S^*

来源：Journal of Physics A-Mathematical and Theoretical, 2012, 45(28): 285001.

DOI：10.1088/1751-8113/45/28/285001

The mathematical properties of the face-centred cubic lattice Green function
G(w) equivalent to 1/pi(3) integral(pi)(0)integral(pi)(0)integral(pi)(0) d theta(1) d theta(2) d theta(3)/w - c(theta(1)) - c(theta(2)) c(theta(3)) - c(theta(3)) c(theta(1)) and the associated logarithmic integral
S(w) equivalent to 1/pi(3) integral(pi)(0)integral(pi)(0)integral(pi)(0) ln[w - c(theta(1)) c(theta(2)) - c(theta(2)) c(theta(3))
c(theta(3)) c(theta(1))]d theta(1) d theta(2) d theta(3)
are investigated, where c(theta) equivalent to cos(theta) 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 F-5(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 F-5(4)
[GRAPHICS]
.
Next the relation dS/dw = G(w), where w is an element of (3,+ infinity), is used to derive the simple formula
G(w) = 1/w {F-2(1) [1/6, 1/3; 1; 27(w + 1)/4w(3)]}(2),
where w lies in a restricted region R-1 of the cut w plane. The limit function
G(w(1)) equivalent to lim(epsilon -> 0+) G(w(1) - i epsilon) = G(R)(w(1)) + iG(I)(w(1))
is also evaluated in the intervals w(1) is an element of (-1, 0] and w(1) is an element of (0, 3]. It is shown that G(R)(w(1)) and G(I)(w(1)) can be expressed in terms of F-2(1)[z(w(1))] hypergeometric functions, where the independent variable z(w(1)) 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.

• 出版日期2012-7-20