A binomial coefficient identity associated with Beukers' conjecture on {A}péry numbers
For a positive integer \(n\), the \(n\)th Apéry number is given by the binomial sum \[ A(n)=\sum_{k=0}^n {n\choose k}^2{n+k\choose k}^2. \] Let \(\alpha(n)\) be determined by the formal power series expansion \[ \sum_{m=1}^{\infty} \alpha(m)q^m=q\prod_{n=1}^{\infty}(1-q^{2n})^4(1-q^{4n})^4= q-4q^3-2q^5+24q^7+\ldots. \] \textit{F. Beukers} [J. Number Theory 25, 201--210 (1987; Zbl 0614.10011)] conjectured that if \(p\) is an odd prime then \(A((p-1)/2)\equiv \alpha(p)\pmod {p^2}\). This conjecture was confirmed by \textit{S. Ahlgren} and \textit{K. Ono} [J. Reine Angew. Math. 518, 187--212 (2000; Zbl 0940.33002)] who showed that it follows from the binomial identity \[ \sum_{k=1}^n {n\choose k}^2{n+k\choose k}^2\Bigl\{1+2kH_{n+k}+2kH_{n-k}- 4kH_k\Bigr\}=0, \] where \(H_k\) is the harmonic number given by \(H_0=0\) and \(H_k= \sum_{j=1}^k 1/j\) for \(k\geq 1\), whose only known proof was nonelementary. In this short note, the author gives an elementary and self-contained proof of a more general identity which implies the above one.
scientific article; zbMATH DE number 2161272