  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.
