Holonomic Gradient Descent for the Fisher-Bingham Distribution on the $n$-dimensional Sphere

We apply the holonomic gradient descent recently introduced in [7] to the maximal likelihood estimate (MLE) with respect to the Fisher-Bingham distribution on the $n$-dimensional sphere. We derive an integrable connection (a Pfaffian system) and a series expansion associated to the normalization constant. These enable us to solve some MLE problems up to $n=7$.