Multivariate Apéry numbers and supercongruences of rational functions

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

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

for primes $p \geq 5$ . 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 $A (n_{1}, n_{2}, n_{3}, n_{4})$ of the rational function

\frac{1}{(1 - x_{1} - x_{2}) (1 - x_{3} - x_{4}) - 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 $λ$ , which also includes the Franel and Yang–Zudilin numbers as well as the Apéry numbers corresponding to $ζ (2)$ . Using the example of the Almkvist–Zudilin numbers, we further indicate evidence of multivariate supercongruences for other Apéry-like sequences.

Keywords

Apéry numbers, supercongruences, diagonals of rational functions, Almkvist–Zudilin numbers

Mathematical Subject Classification 2010

Primary: 11A07

Secondary: 11B83, 11B37, 05A10

Milestones

Received: 7 May 2014

Revised: 10 September 2014

Accepted: 19 October 2014

Published: 28 November 2014

Authors

Armin Straub
Department of Mathematics University of Illinois at Urbana–Champaign Urbana, IL 61801 United States

Vol. 8, No. 8, 2014

Armin Straub

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors