Proof of a supercongruence conjecture of He
The authors prove the following theorem: Theorem 1.2 Let \(p\) be an odd prime such that \(p\equiv 5 \pmod 6\). If \(r\) is a positive integer, then \[ \sum_{k=0}^{tp^r-1} \left(\frac{(\frac{1}{3})_k}{k!}\right)^3 \equiv \begin{cases}\quad p^r\pmod{p^{\min(6,r+3)}},\quad \text{if }r \text{ is even}, \\ -\frac{p^{r+1}}{3} \Gamma_p(\frac{1}{3})^6 \pmod{p^3},\quad \text{if }r \text{ is odd}, \end{cases} \] where \(t=1\) if \(r\) is even, and \(t=2\) if \(r\) is odd. Using \(\binom{z}{k} = (-1) \frac{(-z)_k}{(1)_k}\) they obtain a recent conjectural supercongruence of \textit{Bing He} [Result. Math. 71, No. 3-4, 1223--1234 (2017; Zbl 1421.11005), Conjecture1.3] They use the general method introduced by \textit{L. Long} [Pac. J. Math. 249, No. 2, 405--418 (2011; Zbl 1215.33002)] to prove their results. Furthermore, a suitable hypergeometric series identity and properties of gamma function together with recurrence relations of certain rising factorials are used to deduce the main result.
scientific article; zbMATH DE number 7336787
scientific article