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2 results

  • 4 - A-Hypergeometric Functions

  • Edited by Tom H. Koornwinder, Universiteit van Amsterdam, Jasper V. Stokman, Universiteit van Amsterdam
  • Book:
    Encyclopedia of Special Functions: The Askey-Bateman Project
    Published online:
    30 September 2020
    Print publication:
    15 October 2020, pp 101-121

  • 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.

  • Hypergeometric Polynomials and Integer Programming

  • MUTSUMI SAITO, BERND STURMFELS, NOBUKI TAKAYAMA
  • Journal:
    Compositio Mathematica / Volume 115 / Issue 2 / February 1999
    Published online by Cambridge University Press:
    04 December 2007, pp. 231-240
    Print publication:
    February 1999
    • Article
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  • We examine connections between A-hypergeometric differential equations and the theory of integer programming. In the first part, we develop a ’hypergeometric sensitivity analysis‘ for small variations of constraint constants with creation operators and b-functions. In the second part, we study the indicial polynomial (b-function) along the hyperplane xi=0 via a correspondence between the optimal value of an integer programming problem and the roots of the indicial polynomial. Gröbner bases are used to prove theorems and give counter examples.