Multivariate Apery numbers and supercongruences of rational functions

One of the many remarkable properties of the Apery numbers A(n), introduced in Apery's proof of the irrationality of zeta(3), is that they satisfy the two-term supercongruences A(p(r)m) equivalent to A(p(r-1)m) (mod p(3r)) for primes p >= 5. Similar congruences are conjectured to hold for all Apery-like sequences. We provide a fresh perspective on the supercongruences satisfied by the Apery numbers by showing that they extend to all Taylor coefficients A(n(1), n(2), n(3), n(4)) of the rational function 1/(1 - x(1) - x(2))(1 - x(3) - x(4)) - x(1)x(2)x(3)x(4) The Apery 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 Apery numbers corresponding to zeta(2). Using the example of the Almkvist-Zudilin numbers, we further indicate evidence of multivariate supercongruences for other Apery-like sequences.