Algebraic Independence of an Airy Function, Its Derivative, and Antiderivative

TL;DR

This paper proves that a solution to the Airy differential equation, its derivative, and an antiderivative are algebraically independent over rational functions, using transcendence theory and differential Galois theory.

Contribution

It establishes the algebraic independence of Airy functions, their derivatives, and antiderivatives, combining transcendence and differential Galois theories.

Findings

Proves algebraic independence of Airy function, derivative, and antiderivative

Uses Siegel-Shidlovskii theory of transcendental numbers

Derives results via differential Galois theory

Abstract

Using tools from the Siegel-Shidlovskii theory of transcendental numbers, we prove that a nontrivial solution of the Airy equation, its derivative, and an antiderivative are algebraically independent over the field of rational functions. Courtesy of Michael Singer, the result is also derived from general considerations in differential Galois theory.