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On the cubic lattice Green functions. (English) Zbl 0808.33015

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”.

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
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