Journal of Integer Sequences, Vol. 18 (2015), Article 15.11.7

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


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 Sn 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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(Concerned with sequences A000108 A001448 A001764 A005809 A048990 A176898.)

Received April 30 2015; revised version received December 16 2015. Published in Journal of Integer Sequences, December 16 2015.

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