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

Sums of powers of binomials, their Apéry limits, and Franel's suspicions
Armin Straub, Wadim ZudilinInternational Mathematics Research Notices — Volume 2023, Number 11, 2023, Pages 9861-9879

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.

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BibTeX

@article{franel-aperylimits-2023,
    author = {Armin Straub and Wadim Zudilin},
    title = {Sums of powers of binomials, their Ap\'ery limits, and Franel's suspicions},
    journal = {International Mathematics Research Notices},
    year = {2023},
    volume = {2023},
    number = {11},
    pages = {9861--9879},
    doi = {10.1093/imrn/rnac125},
}