Fractional calculus of Weyl algebra and Fuchsian differential equations

TL;DR

This paper develops a unified fractional calculus framework for linear differential equations, providing new models, explicit solutions, and conditions for irreducibility, especially for Fuchsian equations on the Riemann sphere.

Contribution

It introduces a unified interpretation of various transformations and constructs universal models for Fuchsian equations with specific spectral types.

Findings

Constructed a universal model of Fuchsian equations with given spectral type.

Provided explicit solutions to the connection problem for rigid Fuchsian equations.

Established necessary and sufficient conditions for irreducibility of these equations.

Abstract

We give a unified interpretation of confluences, contiguity relations and Katz's middle convolutions for linear ordinary differential equations with polynomial coefficients and their generalization to partial differential equations. The integral representations and series expansions of their solutions are also within our interpretation. As an application to Fuchsian differential equations on the Riemann sphere, we construct a universal model of Fuchsian differential equations with a given spectral type, in particular, we construct single ordinary differential equations without apparent singularities corresponding to the rigid local systems, whose existence was an open problem presented by Katz. Furthermore we obtain an explicit solution to the connection problem for the rigid Fuchsian differential equations and the necessary and sufficient condition for their irreducibility. We give many examples calculated by our fractional calculus.