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\vskip 1cm{\LARGE\bf
Jacobi Polynomials and Congruences \\
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Involving Some Higher-Order Catalan \\
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Numbers and Binomial Coefficients
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\large
Khodabakhsh Hessami Pilehrood
and
Tatiana Hessami Pilehrood \\
Fields Institute for Research in Mathematical Sciences \\
222 College St, Toronto \\
Ontario M5T 3J1  \\
Canada \\
\href{mailto:hessamik@gmail.com}{\tt hessamik@gmail.com} \\
\href{mailto:hessamit@gmail.com}{\tt hessamit@gmail.com}
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\begin{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 
$S_n=\frac{{6n\choose 3n}{3n\choose 2n}}{2{2n\choose n}(2n+1)},$ 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.
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