On the algebraic dependence of holonomic functions

TL;DR

This paper investigates the algebraic relations between holonomic functions satisfying linear differential equations, providing conditions under which one function is a polynomial in another, with applications to hypergeometric functions and iterated integrals.

Contribution

It establishes criteria linking algebraic dependence of holonomic functions to their differential Galois groups, advancing understanding of their algebraic relations.

Findings

Conditions on differential Galois groups guarantee polynomial dependence.

Applied results to hypergeometric functions.

Extended analysis to iterated integrals.

Abstract

We study the form of possible algebraic relations between functions satisfying linear differential equations. In particular , if f and g satisfy linear differential equations and are algebraically dependent, we give conditions on the differential Galois group associated to f guaranteeing that g is a polynomial in f. We apply this to hypergeometric functions and iterated integrals.