Series expansion and Pfaffian systems of the von Mises distribution on the torus

The normalizing constant of the von Mises distribution on a torus does not have a closed form, where the torus means the direct product of several circles. We derive a series expansion of the normalizing constant together with an error bound, which is useful for numerical computation. We also obtain a Pfaffian system in two-dimensional case, where the Pfaffian system is a set of differential equation necessary to perform the holonomic gradient method.