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# On the Cubic Lattice Green Functions

G. S. Joyce
Proceedings: Mathematical and Physical Sciences
Vol. 445, No. 1924 (May 9, 1994), pp. 463-477
Stable URL: http://www.jstor.org/stable/52609
Page Count: 15
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## Abstract

Wheatstone Physics Laboratory, King's College, University of London, Strand, London WC2R 2LS, U.K. 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 R2 of the η plane, and K (k±) are complete elliptic integrals of the first kind with moduli k± which are given by k$_{\pm}^{2}\equiv$ k±2(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.

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