Staircase polygons, elliptic integrals, Heun functions, and lattice Green functions

Guttmann, A J; Prellberg, Thomas

doi:10.1103/physreve.47.r2233

Cited by 36 publications

(70 citation statements)

References 2 publications

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“…For example, the perimeter generating function for three-dimensional staircase polygons can be expressed in terms of h 6 [18].…”

Section: Key Resultsmentioning

confidence: 99%

Algebraic hypergeometric transformations of modular origin

Maier

2007

Trans. Amer. Math. Soc.

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Abstract. It is shown that Ramanujan's cubic transformation of the Gauss hypergeometric function 2 F 1 arises from a relation between modular curves, namely the covering of X 0 (3) by X 0 (9). In general, when 2 N 7, the N -fold cover of X 0 (N ) by X 0 (N 2 ) gives rise to an algebraic hypergeometric transformation. The N = 2, 3, 4 transformations are arithmetic-geometric mean iterations, but the N = 5, 6, 7 transformations are new. In the final two cases the change of variables is not parametrized by rational functions, since X 0 (6), X 0 (7) are of genus 1. Since their quotients X + 0 (6), X + 0 (7) under the Fricke involution (an Atkin-Lehner involution) are of genus 0, the parametrization is by two-valued algebraic functions. The resulting hypergeometric transformations are closely related to the two-valued modular equations of Fricke and H. Cohn.

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“…For example, the perimeter generating function for three-dimensional staircase polygons can be expressed in terms of h 6 [18].…”

Section: Key Resultsmentioning

confidence: 99%

Algebraic hypergeometric transformations of modular origin

Maier

2007

Trans. Amer. Math. Soc.

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“…In the second we count the pairs of nonintersecting walks regardless of where the two walkers end up. Related results on so-called``vicious walkers" have been obtained by Fisher [3] and others; in particular the numbers of pairs of nonintersecting walks in the d-dimensional lattice have been studied by Guttmann and Prellberg [4].…”

Section: =2nmentioning

confidence: 98%

Counting Pairs of Lattice Paths by Intersections

Gessel

Goddard

Shur

et al. 1996

Journal of Combinatorial Theory, Series A

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“…Green's function for the diamond lattice outside the band is given by (Guttmann and Prellberg, 1993;Ishioka and Koiwa, 1978;Joyce, 1973;Koiwa and Ishioka, 1979):…”

Section: Appendix Amentioning

confidence: 99%