Staircase polygons and recurrent lattice walks (original) (raw)

In this paper we derive a direct relationship between the staircase-polygon-generating function Zd of Guttmann and Prellberg [Phys. Rev. E 47, R2233 (1993)] and the generating function for recurrent lattice walks Pd for the simple (hyper-) cubic lattice in all dimensions d. A recursion formula is obtained for the Zd 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.