On a Class of Hypergeometric Diagonals

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.