Fractional Calculus in C and linear ODE's with Analytic Coefficients, Part 1

TL;DR

This paper develops a new fractional calculus framework in the complex plane, revealing its application to asymptotic analysis of linear ODEs with analytic coefficients, especially for hypergeometric functions.

Contribution

It introduces a novel fractional calculus approach that preserves analyticity for hypergeometric functions and applies it to linear ODE asymptotics.

Findings

Fractional derivatives are analytic in simply connected domains.

Hypergeometric functions retain analyticity under fractional calculus.

Application to asymptotic analysis of linear ODEs with analytic coefficients.

Abstract

The theory of fractional calculus in the complex plane was not built with a specific application in mind. The main obstacle to application was the difficulty with obtaining analytic continuations of fractional derivatives and integrals. It is known that if a function is analytic in a simply connected domain, then its fractional derivatives and integrals are also analytic in this region. In multi-connected domains, in general, this property does not hold. However, for the set of hypergeometric functions $_{p+1}F_{p},p\geq 2$, this property is preserved. We propose a new version of fractional calculus, which allows us to reveal the reason for this preservation, and introduce broad classes of functions for which an appropriate theory can be constructed. This allows us to find a meaningful application of our calculus in the asymptotic theory of linear ODE's with analytic coefficients.