Staircase polygons, elliptic integrals and Heun functions

We discuss the perimeter generating function of $d$-dimensional staircase polygons and relate these to the generating function of the square of the $d$-dimensional multinomial coefficients. These are found to satisfy differential equations of order $(d-1)$. The equations are solved for $d < 5$, and the singularity structure deduced for all values of $d$. The connection with complete elliptic integrals, Heun functions and lattice Green functions is also found.