On two conjectural supercongruences of Apagodu and Zeilberger

Abstract

Let the numbers ${\alpha }_n,{\beta }_n$ and ${\gamma }_n$ denote \begin{align*} α_n=\sum_{k=0}^{n-1}{2k\choose k},\quad β_n=\sum_{k=0}^{n-1}{2k\choose k}\frac{1}{k+1}\quad\text{and}\quad γ_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*} α_{np}&\equiv \left(\frac{p}{3}\right) α_n \pmod{p^2},\\ β_{np}&\equiv \begin{cases} \displaystyle β_n \pmod{p^2},\quad &\text{if $p\equiv 1(\mathrm{mod}3)$},\\ -γ_n \pmod{p^2},\quad &\text{if $p\equiv 2(\mathrm{mod}3)$}, \end{cases} \end{align*} where $\left(\frac{\cdot }{p}\right)$ denotes the Legendre symbol. These two supercongruences were recently conjectured by Apagodu and Zeilberger.

to appear in J. Difference Equ. Appl. This version is a bit different from the final version for publication

Cited by in corpus (2)

• Supercongruences involving dual sequences
• On binomial coefficients modulo squares of primes