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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-η) 1/2 -(1-η) 1/2 ]K(k - ),

where η is a complex variable which lies in a certain region z of the η plain, and K(k ) are complete elliptic integrals of the first kind with moduli k which are given by

k 2 k 2 (η)=1 21 4η(4-η) 1/2 -1 4η(4-η) 1/2 -1 4(2-η)(1-η) 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;α,β,γ,δ;η) 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”.

33E05Elliptic functions and integrals
33E20Functions defined by series and integrals
33E30Functions coming from differential, difference and integral equations
60G50Sums of independent random variables; random walks