Multivariate Ap\'ery numbers and supercongruences of rational functions

One of the many remarkable properties of the Ap\'ery numbers $A (n)$, introduced in Ap\'ery's proof of the irrationality of $\zeta (3)$, is that they satisfy the two-term supercongruences \begin{equation*} A (p^r m) \equiv A (p^{r - 1} m) \pmod{p^{3 r}} \end{equation*} for primes $p \geq 5$. Similar congruences are conjectured to hold for all Ap\'ery-like sequences. We provide a fresh perspective on the supercongruences satisfied by the Ap\'ery numbers by showing that they extend to all Taylor coefficients $A (n_1, n_2, n_3, n_4)$ of the rational function \begin{equation*} \frac{1}{(1 - x_1 - x_2) (1 - x_3 - x_4) - x_1 x_2 x_3 x_4} . \end{equation*} The Ap\'ery 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 Ap\'ery numbers corresponding to $\zeta (2)$. Using the example of the Almkvist--Zudilin numbers, we further indicate evidence of multivariate supercongruences for other Ap\'ery-like sequences.Comment: 19 page

A q-analog of Ljunggren's binomial congruence

We prove a $q$-analog of a classical binomial congruence due to Ljunggren which states that $$\binom{a p}{b p} \equiv \binom{a}{b}$$ modulo $p^3$ for primes $p\ge5$. This congruence subsumes and builds on earlier congruences by Babbage, Wolstenholme and Glaisher for which we recall existing $q$-analogs. Our congruence generalizes an earlier result of Clark.Comment: 6 pages, to be published in the proceedings of FPSAC 201