On the singularities of differential equations satisfied by $E$-functions

TL;DR

The paper investigates the singularities of differential equations associated with $E$-functions and shows that certain algebraic values can be realized without singularities at specific points, answering a question posed by Yves André.

Contribution

It demonstrates that for algebraic values of $E$-functions at algebraic points, there exist $E$-functions with minimal differential equations nonsingular at 1, contrasting with the case at 0 for Bessel functions.

Findings

Existence of $E$-functions with minimal differential equations nonsingular at 1 for given algebraic values.

The property does not hold at 0 for Bessel function values at non-zero algebraic points.

Answers an analogue of a question by Yves André regarding singularities of differential equations for $E$-functions.

Abstract

Let $\xi$ be a value, at an algebraic point, of a Siegel $E$-function. As a special case of a very general interpolation result, we prove that there exists an $E$-function $f$ such that $f(1)=\xi$, and such that 1 is not a singularity of the minimal differential equation satisfied by $f$. We prove that the same property does not hold at the point $0$, when $\xi$ is the value at a non-zero algebraic number of the Bessel function. This answers an analogue of a question asked by Yves Andr{\'e} for $G$-functions.