Jacobi polynomials and congruences involving some higher-order Catalan numbers and binomial coefficients

Abstract

In this paper, we study congruences on sums of products of binomial coefficients that can be proved by using properties of the Jacobi polynomials. We give special attention to polynomial congruences containing Catalan numbers, second-order Catalan numbers, the sequence (\seqnum{A176898}) $S_n=\frac{\left(\genfrac{}{}{0ex}{}{6n}{3n}\right)\left(\genfrac{}{}{0ex}{}{3n}{2n}\right)}{2\left(\genfrac{}{}{0ex}{}{2n}{n}\right)(2n+1)},$ and the binomial coefficients $\left(\genfrac{}{}{0ex}{}{3n}{n}\right)$ and $\left(\genfrac{}{}{0ex}{}{4n}{2n}\right)$. As an application, we address several conjectures of Z.\ W.\ Sun on congruences of sums involving $S_n$ and we prove a cubic residuacity criterion in terms of sums of the binomial coefficients $\left(\genfrac{}{}{0ex}{}{3n}{n}\right)$ conjectured by Z.\ H.\ Sun.