Joyce, G. S. On the cubic lattice Green functions. (English) Zbl 0808.33015 Proc. R. Soc. Lond., Ser. A 445, No. 1924, 463-477 (1994). The author’s abstract: “It is proved that \[ K(k_ +)= [(4- \eta)^{1/2}- (1-\eta)^{1/2}] K(k_ -), \] where \(\eta\) is a complex variable which lies in a certain region \({\mathcal R}_ z\) of the \(\eta\) plain, and \(K(k_ \mp)\) are complete elliptic integrals of the first kind with moduli \(k_ \mp\) which are given by \[ k_ \mp^ 2\equiv k_ \mp^ 2 (\eta)= {\textstyle {1\over 2}} \mp {\textstyle {1\over 4}} \eta(4- \eta)^{1/2}- {\textstyle {1\over 4}} \eta(4-\eta)^{1/2}- {\textstyle {1\over 4}} (2-\eta) (1-\eta)^{1/2}. \] This basic result is then used to express the face-centred cubic and simple cubic lattice Green functions at the origin in terms of the square of a complete elliptic integral of the first kind. Several new identities involving the Heun function \(F(a,b; \alpha,\beta, \gamma,\delta; \eta)\) are also derived. Next it is shown that the three cubic lattice Green functions all have parametric representations which involve the Green function for the two-dimensional honeycomb lattice. Finally, the results are applied to a variety of problems in lattice statistics. In particular, a new simplified formula for the generating function of staircase polygons on a four-dimensional hypercubic lattice is derived”. Reviewer: J.Matkowski (Bielsko-Biała) Cited in 11 Documents MSC: 33E05 Elliptic functions and integrals 33E20 Other functions defined by series and integrals 33E30 Other functions coming from differential, difference and integral equations 60G50 Sums of independent random variables; random walks Keywords:honeycomb lattice; complete elliptic integrals; lattice Green functions; Heun function PDFBibTeX XMLCite \textit{G. S. Joyce}, Proc. R. Soc. Lond., Ser. A 445, No. 1924, 463--477 (1994; Zbl 0808.33015) Full Text: DOI Digital Library of Mathematical Functions: §31.17(ii) Other Applications ‣ §31.17 Physical Applications ‣ Applications ‣ Chapter 31 Heun Functions §31.7(i) Reductions to the Gauss Hypergeometric Function ‣ §31.7 Relations to Other Functions ‣ Properties ‣ Chapter 31 Heun Functions