**Sums of powers of binomials, their Apéry limits, and Franel's suspicions (RISC)**

**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.## Download

Link | Size | Description | Hits |
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2022aperylimits-elpaso.pdf | 1.04 MB | Slides (PDF, 56 pages) | 7 |