Holonomic gradient method for two-way contingency tables

The holonomic gradient method gives an algorithm to efficiently and accurately evaluate normalizing constants and their derivatives. It utilizes holonomic differential equations or holonomic difference equations. Y.Goto and K.Matsumoto gave a system of difference equations for the hypergeometric system of type (k, n). We apply their system to evaluate the normalizing constant and its derivatives of the conditional Poisson or multinomial distribution on two way contingency tables. The modular method in computer algebra has been used for an efficient and exact evaluation. We will also discuss on complexities of these algorithms and their implementation.