The Diagonal and Hadamard Grade of Hypergeometric Functions

Abstract Diagonals of rational functions are an important class of functions arising in number theory, algebraic geometry, combinatorics, and physics. In this paper we study the diagonal grade of a function $f$, which is defined to be the smallest $n$ such that $f$ is the diagonal of a rational function in variables $x_{0},\dots , x_{n}$. We relate the diagonal grade of a function to the nilpotence of the associated differential equation. This allows us to determine the diagonal grade of many hypergeometric functions and answer affirmatively the outstanding question on the existence of functions with diagonal grade greater than $2$. In particular, we show that $ {}_{n}F_{n-1}(\frac{1}{2},\dots , \frac{1}{2};1,\dots ,1 \mid x)$ has diagonal grade $n$ for each $n\geq 1$. Our method also applies to the generating function of the Apéry sequence, which we find to have diagonal grade $3$. We also answer related questions on Hadamard grades posed by Allouche and Mendès France. For example, we show that $ {}_{n}F_{n-1}(\frac{1}{2},\dots , \frac{1}{2};1,\dots ,1 \mid x)$ has Hadamard grade $n$ for all $n\geq 1$.