Singular behaviour of the lattice Green function for thed-dimensional hypercubic lattice

The analytic properties of the d -dimensional hypercubic lattice Green function are investigated, where w = u + i v is a complex variable in the ( u , v ) plane. In particular, the detailed behaviour of G ( d , w ) in the immediate neighbourhood of the branch-point singularities w = ± d is determined. These results are used to derive an asymptotic expansion in powers of 1/ n for the number of random walks on the hypercubic lattice which return to their starting point (not necessarily for the first time) after a walk of 2 n steps. Finally, it is shown that this asymptotic expansion enables one to calculate extremely accurate values for the generalized d -dimensional Watson integral where s > − d /2 and s ≠ 1, 2,....