On a Class of Hypergeometric Diagonals

TL;DR

This paper proves that diagonals of certain algebraic functions are generalized hypergeometric functions, providing explicit parameters and an elementary proof, thus supporting Christol's conjecture for a broad class of functions.

Contribution

It establishes that diagonals of specific algebraic functions are hypergeometric, with explicit parameters, and offers an elementary proof avoiding algorithmic methods.

Findings

Diagonals are generalized hypergeometric functions with explicit parameters.

Supports Christol's conjecture for a large class of algebraic functions.

Provides an elementary, non-algorithmic proof.

Abstract

We prove that the diagonal of any finite product of algebraic functions of the form \begin{align*} {(1-x_1-\dots-x_n)^R}, \qquad R\in\mathbb{Q}, \end{align*} is a generalized hypergeometric function, and we provide an explicit description of its parameters. The particular case $(1-x-y)^R/(1-x-y-z)$ corresponds to the main identity of Abdelaziz, Koutschan and Maillard in [AKM2020, {\S}3.2]. Our result is useful in both directions: on the one hand it shows that Christol's conjecture holds true for a large class of hypergeometric functions, on the other hand it allows for a very explicit and general viewpoint on the diagonals of algebraic functions of the type above. Finally, in contrast to [AKM2020], our proof is completely elementary and does not require any algorithmic help.