Exceptional parameters for generic A-hypergeometric systems

Rank jumps in Codimension 2 A-Hypergeometric Systems

The holonomic rank of the A-hypergeometric system H_A(β) is shown to depend on the parameter vector βwhen the underlying toric ideal I_A is a non Cohen Macaulay codimension 2 toric ideal. The set of exceptional parameters is usually infinite.

The rank of a hypergeometric system

The holonomic rank of the A-hypergeometric system M_A(β) is the degree of the toric ideal I_A for generic parameters; in general, this is only a lower bound. To the semigroup ring of A we attach the ranking arrangement and use this algebraic invariant and the exceptional arrangement of nongeneric parameters to construct a combinatorial formula for the rank jump of M_A(β). As consequences, we obtain a refinement of the stratification of the exceptional arrangement by the rank of M_A(β) and show that the Zariski closure of each of its strata is a union of translates of linear subspaces of the pa…

$A$-Hypergeometric Modules and Gauss--Manin Systems

Let $A$ be a $d$ by $n$ integer matrix. Gel'fand et al. proved that most $A$-hypergeometric systems have an interpretation as a Fourier--Laplace transform of a direct image. The set of parameters for which this happens was later identified by Schulze and Walther as the set of not strongly resonant parameters of $A$. A similar statement relating $A$-hypergeometric systems to exceptional direct images was proved by Reichelt. In this article, we consider a hybrid approach involving neighborhoods $U$ of the torus of $A$ and consider compositions of direct and exceptional direct images. Our main re…

Weyl closure of hypergeometric systems

We show that A-hypergeometric systems and Horn hypergeometric systems are Weyl closed for very generic parameters.

通过基本超几何级数关于参数的$q$-导数得到新的$q$-恒等式

New $q$-identities Via $q$-Derivative of Basic Hypergeometric Series with Respect to Parameters

本文通过$q$-微分和变形$q$-指数算子作用于基本超几何级数，从已知求和与变换公式推导出新的$q$-恒等式。

任意参数超几何函数的代数性

Algebraicity of hypergeometric functions with arbitrary parameters

本研究基于经典交错准则拓展出更一般的设定，对任意参数的广义超几何函数的代数性给出完整分类，并通过多个示例验证了调整后的准则。