GKZ systems with irregular singularities

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We have obtained the following results.(1) The hypergeometric system E_<k, n> is defined on the Grassmanian variety and it has a lot of nice properties. We gave an algorithmic method to derive Kummer type formulas for solutions of the system E_<k, n>. (2) We gave a new method for numerical evaluation of hypergeometric functions in several variables including GKZ hypergeometric functions. Our method translates a given hypergeometric equations into a systems of linear partial differential equations for which standard techniques of numerical analysis can be easily applied. This translation is done by using the Grobner basis method. (3) Slope is a fundamental invariant at irregular singular points. We determined the slopes for GKZ systems associated to monomial curves.