Congruences Related to Dual Sequences and Catalan Numbers (2010.04439v2)

Abstract: During the study of dual sequences, Sun introduced the polynomials [ D_n(x,y)=\sum_{k=0}^{n}{n\choose k}{x\choose k}y^k\text{ and } S_n(x,y)=\sum_{k=0}^{n}\binom{n}{k}\binom{x}{k}\binom{-1-x}{k} y^k. ] Many related congruences have been established and conjectured by Sun. Here we generalize some of them by determining [ \sum_{k=0}^{p-1}D_k(x_1,y_1)D_k(x_2,y_2)\pmod p \text{ and } \sum_{k=0}^{p-1}S_k(x_1,y_1)S_k(x_2,y_2)\pmod p ] for any odd prime $p$ and $p$-adic integers $x_i,\ y_i$ with $i\in{1,2}$. Considering the immediate connection between binomial coefficients and Catalan numbers, we also characterize [ \sum_{n=0}^{p-1}\left(\sum_{k=0}^n {n \choose k} \frac{C_k}{a^k}\right)^2 \pmod {p}, ] where $C_k$ denotes the $k$th Catalan number, $a\in\mathbb{Z}\setminus {0}$ with $\gcd(a,p)=1$. These confirm and generalise some of Sun's conjectures.