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
and the binomial coefficients
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
conjectured by Z. H. Sun.
Full version: pdf,
(Concerned with sequences
Received April 30 2015; revised version received December 16 2015.
Published in Journal of Integer Sequences, December 16 2015.
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