Vol. 8, No. 8, 2014

Download this article
Download this article For screen
For printing
Recent Issues

Volume 10
Issue 9, 1845–2052
Issue 8, 1601–1843
Issue 7, 1373–1600
Issue 6, 1147–1371
Issue 5, 939–1146
Issue 4, 695–938
Issue 3, 451–694
Issue 2, 215–450
Issue 1, 1–214

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
Cover
Editorial Board
Editors' Addresses
Editors' Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Subscriptions
Editorial Login
Contacts
Author Index
To Appear
 
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Multivariate Apéry numbers and supercongruences of rational functions

Armin Straub

Vol. 8 (2014), No. 8, 1985–2008
Abstract

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(prm) A(pr1m)(modp3r)

for primes p 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(n1,n2,n3,n4) of the rational function

1 (1 x1 x2)(1 x3 x4) x1x2x3x4.

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