Multivariate Apéry numbers and supercongruences of rational functions

Abstract

One of the many remarkable properties of the Apéry numbers $A\left(n\right)$, introduced in Apéry’s proof of the irrationality of $\zeta \left(3\right)$, is that they satisfy the two-term supercongruences

$$A\left(p^{r}m\right)\equiv A\left(p^{r-1}m\right)\left(mod{p}^{3r}\right)$$

for primes $p\ge 5$. Similar congruences are conjectured to hold for all Apéry-like sequences. We provide a fresh perspective on the supercongruences satisfied by the Apéry numbers by showing that they extend to all Taylor coefficients $A\left(n_1,n_2,n_3,n_4\right)$ of the rational function

$$\frac{1}{\left(1-x_1-x_2\right)\left(1-x_3-x_4\right)-x_1{x}_2{x}_3{x}_4}.$$

The Apéry numbers are the diagonal coefficients of this function, which is simpler than previously known rational functions with this property.

Our main result offers analogous results for an infinite family of sequences, indexed by partitions $\lambda$, which also includes the Franel and Yang–Zudilin numbers as well as the Apéry numbers corresponding to $\zeta \left(2\right)$. Using the example of the Almkvist–Zudilin numbers, we further indicate evidence of multivariate supercongruences for other Apéry-like sequences.