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

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)A(n) are the famous sequence which underlies Apéry's proof of the irrationality of ζ(3)\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 p5p \geq 5. In this talk, we introduce and discuss a qq-analog of the Apéry numbers. In particular, we prove a supercongruence for these polynomials.

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