Published 2016 ·
The A-hypergeometric differential equations in the present form were introduced by Gel'fand, Zelevinsky, Kapranov  about 30 years ago. Series solutions are multivariable hypergeometric series de ned by a matrix A. Although, there have been analogous approaches before their work, they found that affine toric ideals and their algebraic and combinatorial properties describe solution spaces of the A-hypergeometric differential equations, which also opened new research areas in commutative algebra, combinatorics and algebraic statistics. Several text books describe some topics of these new research areas, see , ,  and their references. The book  and its reference give a comprehensive presentation on the Ahypergeometric equations at the year 2000, and the study has made a substantial progress after it. This chapter hopes to give a directory for these new advances as well as to describe fundamental facts. Applications of A-hypergeometric functions are getting broader. Early applications were mainly for period maps and the algebraic geometry. The interplay with the commutative algebra and combinatorics has been a source of new ideas for both of these and the theory of hypergeometric functions. Recent new applications are for the multivariate analysis in statistics. This chapter starts with systems of differential equations and examples of matrices A which de ne A-hypergeometric functions. We brie y describe an interplay with combinatorics, Gröbner basis, and software systems. Series solutions are discussed with some important examples. In the next to the last section, we illustrate that contiguity relations, isomorphisms, holonomic ranks, reducibility conditions have simple and beautiful descriptions. Recent new applications to statistics will be brie y discussed in the last section.