Talk: On a q-analog of the Apéry numbers (Dalhousie)

On a q-analog of the Apéry numbers (Dalhousie)

Date: 2014/10/18
Occasion: AMS Fall Eastern Sectional Meeting 2014, Special Session on Experimental Mathematics in Number Theory, Analysis, and Combinatorics
Place: Dalhousie University


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, \[ A (p^r m) \equiv A (p^{r - 1} m) \pmod{p^{3 r}} \] 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.


406.85 KB Slides (PDF, 44 pages) 1149