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Jacobi Polynomials and Congruences
Involving Some Higher-Order Catalan
Numbers and Binomial Coefficients Khodabakhsh Hessami Pilehrood and Tatiana Hessami Pilehrood
Fields Institute for Research in Mathematical Sciences
222 College St, Toronto
Ontario M5T 3J1
Canada
mailto:hessamik@gmail.comhessamik@gmail.com
mailto:hessamit@gmail.comhessamit@gmail.com

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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.