Proof of Two Supercongruences of Guillera and Zudilin

Abstract

In $2012$, Guillera and Zudilin established the following two supercongruences involving truncated Ramanujan-type series: for any odd prime $p>2$, \begin{align*} \sum_{n=0}^{p-1}\frac{(\frac{1}{2})_n(\frac{1}{3})_n(\frac{1}{4})_n(\frac{3}{4})_n}{(1)_n^5}(-1)^n\left(172n^2+75n+9\right)\left(\frac{27}{16}\right)^n\equiv 9p^2 \pmod{p^5}, \end{align*} and \begin{align*} \sum_{n=0}^{p-1}\frac{(\frac{1}{2})_n(\frac{1}{3})_n(\frac{2}{3})_n}{(1)_n^3}\left(11n+3\right)\left(\frac{27}{16}\right)^n\equiv 3p \pmod{p^3}, \end{align*} where $(a)_n=\prod_{k=0}^{n-1}(a+k)$ denotes the Pochhammer symbol (rising factorial). In this paper, we mainly apply the Wilf-Zeilberger (WZ) method and symbolic summation techniques to prove these two supercongruences.