On two conjectural supercongruences of Apagodu and Zeilberger

Abstract

Let the numbers $\alpha_n,\beta_n$ and $\gamma_n$ denote \begin{align*} \alpha_n=\sum_{k=0}^{n-1}{2k\choose k},\quad \beta_n=\sum_{k=0}^{n-1}{2k\choose k}\frac{1}{k+1}\quad\text{and}\quad \gamma_n=\sum_{k=0}^{n-1}{2k\choose k}\frac{3k+2}{k+1}, \end{align*} respectively. We prove that for any prime $p\ge 5$ and positive integer $n$ \begin{align*} \alpha_{np}&\equiv \left(\frac{p}{3}\right) \alpha_n \pmod{p^2},\\ \beta_{np}&\equiv \begin{cases} \displaystyle \beta_n \pmod{p^2},\quad &\text{if $p\equiv 1\pmod{3}$},\\ -\gamma_n \pmod{p^2},\quad &\text{if $p\equiv 2\pmod{3}$}, \end{cases} \end{align*} where $\left(\frac{\cdot}{p}\right)$ denotes the Legendre symbol. These two supercongruences were recently conjectured by Apagodu and Zeilberger.