Sums of powers of binomials, their Apéry limits, and Franel's suspicions
arXiv:2112.09576 · doi:10.1093/imrn/rnac125
Abstract
We explicitly determine the Apéry limits for the 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. Namely, we prove the conjectured minimal order under the assumption that such a recurrence can be obtained via creative telescoping.
19 pages