The mathematical properties of the lattice Green function
[graphic not available: see fulltext] are investigated, where w=w1+iw2 lies in a complex plane which is cut from w=−1 to w=3, and {ℓ1,ℓ2,ℓ3} is a set of integers with ℓ1+ℓ2+ℓ3 equal to an even integer. In particular, it is proved that G(2n,0,0;w), where n=0,1,2,…, is a solution of a fourth-order linear differential equation of the Fuchsian type with four regular singular points at w=−1,0,3 and ∞. It is also shown that G(2n,0,0;w) satisfies a five-term recurrence relation with respect to the integer variable n. The limiting function
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\begin{document}$$G^{-}(2n,0,0;w_1)\equiv\lim_{\epsilon\rightarrow0+}G(2n,0,0;w_1-\mathrm{i}\epsilon) =G_{\mathrm{R}}(2n,0,0;w_1)+\mathrm{i}G_{\mathrm {I}}(2n,0,0;w_1) ,\nonumber $$\end{document} where w1∈(−1,3), is evaluated exactly in terms of 2F1 hypergeometric functions and the special cases G−(2n,0,0;0), G−(2n,0,0;1) and G(2n,0,0;3) are analysed using singular value theory. More generally, it is demonstrated that G(ℓ1,ℓ2,ℓ3;w) can be written in the form
[graphic not available: see fulltext] where [inline-graphic not available: see fulltext] are rational functions of the variable ξ, K(k−) and E(k−) are complete elliptic integrals of the first and second kind, respectively, with
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\begin{document}$$k_{-}^2\equiv k_{-}^2(w)={1\over2}- {2\over w} \biggl(1+{1\over w} \biggr)^{-{3\over2}}- {1\over2} \biggl(1-{1\over w} \biggr ) \biggl(1+{1\over w} \biggr)^{-{3\over2}} \biggl(1-{3\over w} \biggr)^{1\over2}\nonumber $$\end{document} and the parameter ξ is defined as
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\begin{document}$$\xi\equiv\xi(w)= \biggl(1+\sqrt{1-{3\over w}} \,\biggr)^{-1} \biggl(-1+\sqrt{1+{1\over w}} \,\biggr) .\nonumber $$\end{document} This result is valid for all values of w which lie in the cut plane. The asymptotic behaviour of G−(2n,0,0;w1) and G(2n,0,0;w1) as n→∞ is also determined. In the final section of the paper a new 2F1 product form for the anisotropic face-centred cubic lattice Green function is given.
