On Christol's conjecture

TL;DR

This paper demonstrates that several unresolved hypergeometric functions related to Christol's conjecture are diagonals of rational functions, providing new evidence and arguments supporting the conjecture's validity.

Contribution

The authors prove that specific unresolved $_3F_2$ and $_4F_3$ hypergeometric functions are diagonals of rational functions, advancing understanding of Christol's conjecture.

Findings

Certain $_3F_2$ functions are diagonals of rational functions.

Unresolved $_4F_3$ examples are also diagonals.

Evidence suggests a specific $_3F_2$ function is likely a diagonal.

Abstract

We show that the unresolved examples of Christol's conjecture $ \, _3F_{2}\left([2/9,5/9,8/9],[2/3,1],x\right)$ and $_3F_{2}\left([1/9,4/9,7/9],[1/3,1],x\right)$, are indeed diagonals of rational functions. We also show that other $\, _3F_2$ and $\, _4F_3$ unresolved examples of Christol's conjecture are diagonals of rational functions. Finally we give two arguments that show that it is likely that the $\, _3F_2([1/9, 4/9, 5/9], \, [1/3,1], \, 27 \cdot x)$ function is a diagonal of a rational function.