arminstraub.com

# Talk: Sums of powers of binomials, their Apéry limits, and Franel's suspicions (El Paso)

Sums of powers of binomials, their Apéry limits, and Franel's suspicions (El Paso)

Date: 2022/09/18
Occasion: AMS Fall Central Sectional Meeting, Special Session on The Intersection of Number Theory and Combinatorics
Place: University of Texas at El Paso

## Abstract

Apéry's proof of the irrationality of $$\zeta(3)$$ relies on representing that value as the limit of the quotient of two rational solutions to a three-term recurrence. We review such Apéry limits and explore a particularly simple instance. We then explicitly determine the Apéry limits attached to sums of powers of binomial coefficients. As an application, we prove a weak version of Franel's conjecture on the order of the recurrences for these sequences. This is based on joint work with Wadim Zudilin.