  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})
$S_n=\frac{{6n\choose 3n}{3n\choose 2n}}{2{2n\choose n}(2n+1)},$ and the
binomial coefficients ${3n\choose n}$ and ${4n\choose 2n}$. 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 ${3n\choose n}$ conjectured by Z.\ H.\ Sun.
