  The holonomic gradient method gives an algorithm to efficiently and
accurately evaluate normalizing constants and their derivatives. We apply the
holonomic gradient method in the case of the conditional Poisson or multinomial
distribution on two way contingency tables. We utilize the modular method in
computer algebra for an efficient and exact evaluation, and we discuss on
complexities of these algorithms and their implementation. We also discuss on a
theoretical aspect of the distribution from the viewpoint of the conditional
maximum likelihood estimation.
