Proof of Two Supercongruences of Guillera and Zudilin

TL;DR

This paper proves two supercongruences involving truncated Ramanujan-type series, originally established by Guillera and Zudilin in 2012, using the Wilf-Zeilberger method and symbolic summation.

Contribution

It provides rigorous proofs of two supercongruences using WZ and symbolic summation, confirming their validity.

Findings

Confirmed supercongruences modulo prime powers for specific series.

Applied WZ method and symbolic summation to establish supercongruences.

Validated conjectures posed by Guillera and Zudilin in 2012.

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.