Congruences Related to Dual Sequences and Catalan Numbers

During the study of dual sequences, Sun introduced the polynomials $$D_n(x,y)=\sum _{k=0}^n\left(\genfrac{}{}{0px}{}{n}{k}\right)\left(\genfrac{}{}{0px}{}{x}{k}\right)y^k\text{ and }{S}_n(x,y)=\sum _{k=0}^n\left(\genfrac{}{}{0px}{}{n}{k}\right)\left(\genfrac{}{}{0px}{}{x}{k}\right)\left(\genfrac{}{}{0px}{}{-1-x}{k}\right)y^k\mathrm{.}$$ Many related congruences have been established and conjectured by Sun. Here we generalize some of them by determining $$\sum _{k=0}^{p-1}{D}_k(x_1,y_1)D_k(x_2,y_2)(\mathrm{m}\mathrm{o}\mathrm{d}p)\text{ and }\sum _{k=0}^{p-1}{S}_k(x_1,y_1)S_k(x_2,y_2)(\mathrm{m}\mathrm{o}\mathrm{d}p)$$ 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 $$\sum _{n=0}^{p-1}{\left(\sum _{k=0}^n\left(\genfrac{}{}{0px}{}{n}{k}\right)\frac{C_k}{a^k}\right)}^2(\mathrm{m}\mathrm{o}\mathrm{d}p),$$ 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.