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

We show that the generating function for d-dimensional staircase polygons (by perimeter) can be expressed in terms of the generating function for the square of d-dimensional multinomial coefficients. This latter generating function is found to satisfy a linear, homogeneous differential equation of order d-1. This equation is solved for d{≤}4. For d=3 and d=4 the solution is obtained in terms of Heun functions, which are then shown to be expressible in terms of the complete elliptic integral of the first kind. The solutions are also shown to be related to lattice Green functions on three-dimensional lattices. The critical behavior of this model is determined exactly in all dimensions.