Staircase polygons and recurrent lattice walks

M. L. Glasser; E. Montaldi

doi:10.1103/PhysRevE.48.R2339

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Staircase polygons and recurrent lattice walks

M. L. Glasser

E. Montaldi

Department of Physics and Clarkson Institute for Statistical Physics, Clarkson University, Potsdam, New York 13699-5820

Dipartimento di Fisica, Università di Milano, Via Celoria 16, 20133 Milano, Italy

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Phys. Rev. E 48, R2339(R) – Published 1 October, 1993

DOI: https://doi.org/10.1103/PhysRevE.48.R2339

Abstract

In this paper we derive a direct relationship between the staircase-polygon-generating function $Z_{d}$ of Guttmann and Prellberg [Phys. Rev. E 47, R2233 (1993)] and the generating function for recurrent lattice walks $P_{d}$ for the simple (hyper-) cubic lattice in all dimensions d. A recursion formula is obtained for the $Z_{d}$ with respect to dimension, which leads to a simplified derivation of Guttmann and Prellberg’s result for d=3, avoiding the use of the Heun function, and a derivation of their formula for d=4 from an integral representation is given in the Appendix.

References (5)

A. J. Guttmann and T. Prellberg, Phys. Rev. E 47, R2233 (1993).
G. S. Joyce, Philos. Trans. R. Soc. London 273, 583 (1973).
E. T. Whittaker and G. N. Watson, A Course in Modern Analysis, 4th ed. (Cambridge University Press, Cambridge, 1963), p. 229.
L. J. Slater, Generalized Hypergeometric Functions (Cambridge University Press, Cambridge, 1966), p. 62, Eq. (2.4.1.2).
W. N. Bailey, Generalized Hypergeometric Series (Cambridge University Press, Cambridge, 1935), p. 81.

[c1] A. J. Guttmann and T. Prellberg, Phys. Rev. E 47, R2233 (1993).

[c2] G. S. Joyce, Philos. Trans. R. Soc. London 273, 583 (1973).

[c3] E. T. Whittaker and G. N. Watson, A Course in Modern Analysis, 4th ed. (Cambridge University Press, Cambridge, 1963), p. 229.

[c4] L. J. Slater, Generalized Hypergeometric Functions (Cambridge University Press, Cambridge, 1966), p. 62, Eq. (2.4.1.2).

[c5] W. N. Bailey, Generalized Hypergeometric Series (Cambridge University Press, Cambridge, 1935), p. 81.

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Staircase polygons and recurrent lattice walks

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