Rank Jumps in Codimension 2A-hypergeometric Systems

Rank Jumps in Codimension

Homological Methods for Hypergeometric Families

We analyze the behavior of the holonomic rank in families of holonomicsystems over complex algebraic varieties by providing homological criteria for rank-jumpsin this general setting. Then we investigate rank-jump behavior for hypergeometric sys-tems HA(β) arising from a d × n integer matrix A and a parameter β ∈ C. To doso we introduce an Euler–Koszul functor for hypergeometric...

Combinatorics of Rank Jumps in Simplicial Hypergeometric Systems

Let A be an integer d × n matrix, and assume that the convex hull conv(A) of its columns is a simplex of dimension d− 1 not containing the origin. It is known that the semigroup ring C[NA] is Cohen–Macaulay if and only if the rank of the GKZ hypergeometric system HA(β) equals the normalized volume of conv(A) for all complex parameters β ∈ Cd (Saito, 2002). Our refinement here shows that HA(β) h...

ON AN EXTENSION OF A QUADRATIC TRANSFORMATION FORMULA DUE TO GAUSS

The aim of this research note is to prove the following new transformation formula begin{equation*} (1-x)^{-2a},_{3}F_{2}left[begin{array}{ccccc} a, & a+frac{1}{2}, & d+1 & & \ & & & ; & -frac{4x}{(1-x)^{2}} \ & c+1, & d & & end{array}right] \ =,_{4}F_{3}left[begin{array}{cccccc} 2a, & 2a-c, & a-A+1, & a+A+1 & & \ & & & & ; & -x \ & c+1, & a-A, & a+A & & end{array} right], end{equation*} wher...

Se p 20 03 BIVARIATE HYPERGEOMETRIC D - MODULES

We undertake the study of bivariate Horn systems for generic parameters. We prove that these hypergeometric systems are regular holonomic, and we provide an explicit formula for their holonomic rank as well as bases of their spaces of complex holomorphic solutions. We also obtain analogous results for the generalized hypergeometric systems arising from lattices of any rank.