Gamma structures and Gauss's contiguity

TL;DR

This paper introduces gamma structures on hypergeometric D-modules, linking their monodromy to gamma products and hypergeometric exponents, with explicit computations of monodromy matrices.

Contribution

It defines gamma structures on hypergeometric D-modules and provides explicit formulas for their monodromy in terms of algebraic and matrix data.

Findings

Gamma structures are introduced on hypergeometric D-modules.

Monodromy can be expressed algebraically via hypergeometric exponents.

Explicit formulas for hypergeometric monodromy matrices are derived.

Abstract

We introduce gamma structures on regular hypergeometric D--modules in dimension 1 as special one--parametric systems of solutions on the compact subtorus. We note that a balanced gamma product is in the Paley--Wiener class and show that the monodromy with respect to the gamma structure is expressed algebraically in terms of the hypergeometric exponents. We compute the hypergeometric monodromy explicitly in terms of certain diagonal matrices, Vandermonde matrices and their inverses (or generalizations of those in the resonant case).