On a class of hypergeometric diagonals

Abstract

We prove that the diagonal of any finite product of algebraic functions of the form ( 1 − x 1 − ⋯ − x n ) R , R ∈ Q , \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 ) (1-x-y)^R/(1-x-y-z) corresponds to the main identity of Abdelaziz, Koutschan and Maillard in [J. Phys. A 53 (2020), 205201, 16 pp., §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 the approach of Abdelaziz, Koutschan and Maillard, our proof is completely elementary and does not require any algorithmic help.