Super congruences concerning binomial coefficients and Apéry-like numbers

Abstract

Let $p$ be a prime with $p>3$, and let $a,b$ be two rational $p-$integers. In this paper we present general congruences for ${\sum }_{k=0}^{p-1}\left(\genfrac{}{}{0ex}{}{a}{k}\right)\left(\genfrac{}{}{0ex}{}{-1-a}{k}\right)\frac{p}{k+b}(\mathrm{mod}{p}^2)$. For $n=0,1,2,\dots$ let $D_n$ and $b_n$ be Domb and Almkvist-Zudilin numbers, respectively. We also establish congruences for $$\sum _{n=0}^{p-1}\frac{D_n}{16^n},\sum _{n=0}^{p-1}\frac{D_n}{4^n},\sum _{n=0}^{p-1}\frac{b_n}{(-3{)}^n},\sum _{n=0}^{p-1}\frac{b_n}{(-27{)}^n}(\mathrm{mod}{p}^2)$$ in terms of certain binary quadratic forms.

35 pages

Cited by in corpus (2)

• Congruences for two types of Apery-like sequences
• A $p$-adic analogue of Chan and Verrill's formula for $1/\pi$