Middle Laplace transform and middle convolution for linear Pfaffian systems with irregular singularities

TL;DR

This paper introduces the middle Laplace transform for linear Pfaffian systems, providing a new algebraic and categorical framework, and extends the concept of middle convolution to systems with irregular singularities.

Contribution

It defines the middle Laplace transform with fundamental properties and generalizes Haraoka's middle convolution to irregular singularities, supported by examples involving hypergeometric functions.

Findings

Middle Laplace transform is invertible and irreducible.

Middle convolution is generalized to systems with irregular singularities.

Examples include hypergeometric functions with two variables.

Abstract

We introduce a transformation of linear Pfaffian systems, which we call the middle Laplace transform, as a formulation of the Laplace transform from the perspective of Katz theory. While the definition of the middle Laplace transform is purely algebraic, its categorical interpretation is also provided. We then show the fundamental properties (invertibility, irreducibility) of the middle Laplace transform. As an application of the middle Laplace transform, we define the middle convolution for linear Pfaffian systems with irregular singularities. This gives a generalization of Haraoka's middle convolution, which was defined for linear Pfaffian systems with logarithmic singularities. The fundamental properties (additivity, irreducibility) of the middle convolution follow from the properties of the middle Laplace transform. Some examples related to hypergeometric functions with two variables are also given.