Congruences Related to Dual Sequences and Catalan Numbers

TL;DR

This paper generalizes and proves new congruences involving dual sequences, Catalan numbers, and binomial coefficients modulo primes, confirming and extending several of Sun's conjectures.

Contribution

It extends Sun's work by deriving new congruences for dual sequences and Catalan-related sums modulo primes, providing a broader understanding of their number-theoretic properties.

Findings

Derived congruences for sums involving dual sequences modulo p

Characterized sums of Catalan numbers modulo p

Confirmed and generalized several of Sun's conjectures

Abstract

During the study of dual sequences, Sun introduced the polynomials \[ D_n(x,y)=\sum_{k=0}^{n}{n\choose k}{x\choose k}y^k\text{ and } S_n(x,y)=\sum_{k=0}^{n}\binom{n}{k}\binom{x}{k}\binom{-1-x}{k} y^k. \] Many related congruences have been established and conjectured by Sun. Here we generalize some of them by determining \[ \sum_{k=0}^{p-1}D_k(x_1,y_1)D_k(x_2,y_2)\pmod p \text{ and } \sum_{k=0}^{p-1}S_k(x_1,y_1)S_k(x_2,y_2)\pmod p \] for any odd prime $p$ and $p$-adic integers $x_i,\ y_i$ with $i\in\{1,2\}$. Considering the immediate connection between binomial coefficients and Catalan numbers, we also characterize \[ \sum_{n=0}^{p-1}\left(\sum_{k=0}^n {n \choose k} \frac{C_k}{a^k}\right)^2 \pmod {p}, \] where $C_k$ denotes the $k$th Catalan number, $a\in\mathbb{Z}\setminus \{0\}$ with $\gcd(a,p)=1$. These confirm and generalise some of Sun's conjectures.