Abstract
This chapter introduces the A-hypergeometric system of linear partial differential equations. It presents the known results of the A-hypergeometric system with a two-row matrix A and explains its relationship with integer partitions. The A-hypergeometric distribution is a class of discrete exponential families whose normalization constant is the A-hypergeometric polynomial. The A-hypergeometric distribution emerges in multinomial sampling of log-affine models and is conditional on sufficient statistics. After presenting the properties of the A-hypergeometric distribution, the chapter discusses the maximum likelihood estimation of the A-hypergeometric distribution of the two-row matrix A. Especially, using the properties of partition polytopes, it proves the nonexistence theorem of the maximum likelihood estimator. Finally, it introduces holonomic gradient methods (HGMs), which numerically solve holonomic systems without combinatorial enumeration, and applies a difference HGM to the A-hypergeometric polynomials of a two-row matrix A.
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