**Supercongruences for polynomial analogs of the Apéry numbers (Integers 2018)**

**Date:** 2018/10/05
**Occasion:** Integers Conference 2018
**Place:** University of Augusta

## Abstract

The Apéry numbers \(A(n)\) are the famous sequence which underlies Apéry's proof of the irrationality of \(\zeta(3)\). Together with their siblings, known as Apéry-like, they enjoy remarkable properties, including connections with modular forms, and have appeared in various contexts. One of their (still partially conjectural) properties is that these sequences satisfy supercongruences, a term coined by Beukers to indicate that the congruences are modulo exceptionally high powers of primes. For instance, \begin{equation*} A (p^r m) \equiv A (p^{r - 1} m) \pmod{p^{3 r}} \end{equation*} for primes \(p \geq 5\). In this talk, we introduce and discuss a \(q\)-analog of the Apéry numbers. In particular, we prove a supercongruence for these polynomials.## Download

Link | Size | Description | Hits |
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2018qapery-integers.pdf | 397.02 KB | Slides (PDF, 41 pages) | 1219 |