Proof of a Dwork-type supercongruence by induction

Abstract

In this paper we prove a Dwork-type supercongruence: for any prime $ p\geq3 $ and integer $ r\geq 1 $, $ \begin{align*} \sum\limits_{k = 0}^{p^r-1}\frac{3k+1}{16^k}{\binom{2k}{k}}^3\equiv p\sum\limits_{k = 0}^{p^{r-1}-1}\frac{3k+1}{16^k}{\binom{2k}{k}}^3\pmod{p^{3r+1-\delta_{p, 3}}}, \end{align*} $ which extends a result of Guo and Zudilin.

Keywords

• supercongruences
• dwork-type supercongruences
• generalized harmonic numbers of order m
• induction
• jacobsthal's binomial congruence