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Register a new userModified A-hypergeometric Systems
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Authors
Nobuki Takayama
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We will introduce a modified system of A-hypergeometric system (GKZ system) by applying a change of variables for Groebner deformations and study its Groebner basis and the indicial polynomials along the exceptional hypersurface.
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Gauss hypergeometric functions with a dihedral monodromy group can be expressed as elementary functions, since their hypergeometric equations can be transformed to Fuchsian equations with cyclic monodromy groups by a quadratic change of the argument variable. The paper presents general elementary expressions of these dihedral hypergeometric functions, involving finite bivariate sums expressible as terminating Appells F2 or F3 series. Additionally, trigonometric expressions for the dihedral functions are presented, and degenerate cases (logarithmic, or with the monodromy group Z/2Z) are considered.
This is the third revision. We study bases of Pfaffian systems for $A$-hypergeometric system. Grobner deformations give bases. These bases also give those for twisted cohomology groups. For hypergeometric system associated to a class of order polytopes, these bases have a combinatorial description. The size of the bases associated to a subclass of the order polytopes have the growth rate of the polynomial order. Bases associated to two chain posets and bouquets are studied.
We give a bijection between a quotient space of the parameters and the space of moments for any $A$-hypergeometric distribution. An algorithmic method to compute the inverse image of the map is proposed utilizing the holonomic gradient method and an asymptotic equivalence of the map and the iterative proportional scaling. The algorithm gives a method to solve a conditional maximum likelihood estimation problem in statistics. Our interplay between the theory of hypergeometric functions and statistics gives some new formulas of $A$-hypergeometric polynomials.
This is a brief overview of the status of the theory of elliptic hypergeometric functions to the end of 2012 written as a complementary chapter to the Russian edition of the book by G.E. Andrews, R. Askey, and R. Roy, Special Functions, Encycl. of Math. Appl. 71, Cambridge Univ. Press, 1999.
A special singular limit $omega_1/omega_2to 1$ is considered for the Faddeev modular quantum dilogarithm (hyperbolic gamma function) and corresponding hyperbolic integrals. It brings a new class of hypergeometric identities associated with bilateral sums of Mellin-Barnes type integrals of particular Pochhammer symbol products.
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