Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0808.33015
Joyce, G.S.
On the cubic lattice Green functions. (English)
[J] Proc. R. Soc. Lond., Ser. A 445, No.1924, 463-477 (1994). ISSN 0080-4630

The author's abstract: ``It is proved that $$K(k\sb +)= [(4- \eta)\sp{1/2}- (1-\eta)\sp{1/2}] K(k\sb -),$$ where $\eta$ is a complex variable which lies in a certain region ${\cal R}\sb z$ of the $\eta$ plain, and $K(k\sb \mp)$ are complete elliptic integrals of the first kind with moduli $k\sb \mp$ which are given by $$k\sb \mp\sp 2\equiv k\sb \mp\sp 2 (\eta)= {\textstyle {1\over 2}} \mp {\textstyle {1\over 4}} \eta(4- \eta)\sp{1/2}- {\textstyle {1\over 4}} \eta(4-\eta)\sp{1/2}- {\textstyle {1\over 4}} (2-\eta) (1-\eta)\sp{1/2}.$$ 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''.
[J.Matkowski (Bielsko-Biała)]

MSC 2000:: *33E05 Elliptic functions and integrals
33E20 Functions defined by series and integrals
33E30 Functions coming from diff., difference and integral equations
60G50 Sums of independent random variables

Keywords: honeycomb lattice; complete elliptic integrals; lattice Green functions; Heun function

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]