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

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

Date: 2022/03/13
Occasion: Southern Regional Number Theory Conference
Place: Louisiana State University

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.

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