Sums of powers of binomials, their Ap\'ery limits, and Franel's suspicions

TL;DR

This paper explicitly computes Apéry limits for sums of binomial powers and demonstrates a weak form of Franel's conjecture regarding the minimal order of related recurrences, assuming they are derived through creative telescoping.

Contribution

It provides explicit Apéry limits for binomial power sums and proves a weak version of Franel's conjecture on recurrence orders under specific assumptions.

Findings

Explicit Apéry limits for sums of powers of binomials.

A proof of the minimal recurrence order under the creative telescoping assumption.

Confirmation of a weak version of Franel's conjecture.

Abstract

We explicitly determine the Ap\'ery 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.