Jacobi polynomials and congruences involving some higher-order Catalan numbers and binomial coefficients

TL;DR

This paper investigates congruences involving binomial coefficients, Catalan numbers, and Jacobi polynomials, proving several conjectures and establishing new criteria related to cubic residuacity.

Contribution

It introduces novel congruence results involving higher-order Catalan numbers and binomial coefficients using properties of Jacobi polynomials, addressing conjectures by Z. W. Sun.

Findings

Proved conjectures on sums involving higher-order Catalan numbers.

Established a cubic residuacity criterion based on binomial coefficient sums.

Derived new polynomial congruences involving Jacobi polynomials.

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 (\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.