4 - A-Hypergeometric Functions

Summary

The A-hypergeometric or GKZ hypergeometric system of differential equations in the present form were introduced by Gel'fand, Zelevinsky, and Kapranov about 30 years ago. Series solutions are multivariable hypergeometric series defined by a matrix A. 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, polyhedral geometry, and algebraic statistics. This chapter describes fundamental facts about the system and its solutions, and also gives pointers to recent advances. Applications of A-hypergeometric functions are getting broader. Early applications were mainly to period maps and algebraic geometry. The interplay with commutative algebra and combinatorics has been a source of new ideas for these two fields and for the theory of hypergeometric functions. Recent new applications are to multivariate analysis in statistics.