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arXiv · 2112.09576

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

Armin Straub·Wadim Zudilin
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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.

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Armin Straub, Wadim Zudilin. 2021-12-17. Sums of powers of binomials, their Ap\'ery limits, and Franel's suspicions. https://doi.org/10.1093/imrn%2Frnac125

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