G S Joyce and R T Delves 2004 J. Phys. A: Math. Gen. 37 3645 doi:10.1088/0305-4470/37/11/008
Exact product forms for the simple cubic lattice Green function: I
G S Joyce and R T Delves
Show affiliationsThe analytical properties of the lattice Green function ![\fl G(n,n,n;w)={1\over \pi^3}\int_0^{\pi}\!\!\!\int_0^{\pi}\!\!\!\int_0^{\pi}{{\cos{n}\theta{_1}\cos{n}\theta{_2}\cos{n}\theta{_3}}\over{w-\cos \theta_1-\cos \theta_2-\cos \theta_3}}\, {\rm d}\theta_1\, {\rm d}\theta_2\, {\rm d}\theta_3\nonumber\\[-7pt]](http://ej.iop.org/images/0305-4470/37/11/008/jpa171401ueq01.gif)
are investigated, where n is an integer and w is a complex variable. In particular, it is demonstrated that G(n, n, n; w) is a solution of a fourth-order linear differential equation of the Fuchsian type. From this differential equation it is found that G(n, n, n; w) can be evaluated in terms of a product of two Heun functions {Hj(n, v):j = 1, 2}, where ![v\equiv v(w) = {1\over{w^2}}\left(1+\sqrt{1-{1\over w^2}}\, \right)^{-1}\left(1+\sqrt{1-{9\over w^2}} \,\right)^{-1}.\nonumber\\[-6pt]](http://ej.iop.org/images/0305-4470/37/11/008/jpa171401ueq02.gif)
A detailed discussion of the properties of {Hj(n, v):j = 1, 2} is then given. The Heun function results are used to prove that the product form for G(n, n, n; w) can be expressed in terms of complete elliptic integrals of the first and second kinds. It is also shown that G(n, n, n; w) can be written in the hypergeometric form ![\fl wG(n,n,n;w)={{(3n)!}\over{(3^n n!)^3}}\left[{w\over 3}\left(1-\sqrt{1-{9\over w^2}} \,\right)\right]^{3n}{}_2F_1\left({1\over 3},{2\over 3};n+1;\eta_{+}\right)\\
\times\, {}_2F_1\left({1\over 3},{2\over 3};n+1;\eta_{-}\right)](http://ej.iop.org/images/0305-4470/37/11/008/jpa171401ueq03.gif)
where ![\fl \eta_{\pm} \equiv \eta_{\pm}(w) = {1\over{8w^2}}\left[4w^2+(9-4w^2)\sqrt{1-{9\over{w^2}}}\pm 27\sqrt{1-{1\over{w^2}}} \,\right].](http://ej.iop.org/images/0305-4470/37/11/008/jpa171401ueq04.gif)
This formula is valid for varying values of w in the neighbourhood of w = ∞, provided that the argument function η+(w) does not take real values in the interval (1, + ∞). Finally, this 2F1 product form is used to determine the asymptotic behaviour of G(n, n, n; w) as n → ∞.
- PACS
- MSC
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82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs
- Subjects
- Dates
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Issue 11 (19 March 2004)
Received 6 November 2003
Published 2 March 2004
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Exact product forms for the simple cubic lattice Green function: I
G S Joyce and R T Delves 2004 J. Phys. A: Math. Gen. 37 3645
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