Seminar talk: ?Sums of powers of binomials, their Apery limits, and Franel?s suspicions? (Speaker: Prof. Armin Straub, University of South Alabama)
Sprache des Titels:
Ap\'ery'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 Ap\'ery limits and explore a particularly simple instance. We then explicitly determine the Ap\'ery 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.