[Paper Review] Various congruences involving binomial coefficients and higher-order Catalan numbers

Zhi‐Wei Sun|ArXiv.org|Sep 21, 2009

Advanced Mathematical Identities11 references10 citations

TL;DR

This paper establishes new congruences modulo prime powers for weighted sums of binomial coefficients and higher-order Catalan numbers, particularly focusing on $Σ_{k=0}^{p^a-1} \binom{3k}{k}/m^k$ and $\sum_{k=0}^{p^a-1} C_k^{(3)}/5^k$ modulo $p$. Using third- and fourth-order linear recurrences, cubic residues, and Jacobi symbols, it derives exact evaluations involving Legendre symbols and recurrence sequences, extending prior work on generalized Catalan numbers and binomial sums modulo $p$. The key result is a closed-form expression for $\sum C_k^{(3)}/5^k \equiv 1 \pmod{p}$ and $\sum \bar{C}_k^{(3)}/5^k \equiv 3 \pmod{p}$, with explicit formulas depending on $\left(\frac{-2}{p^a}\right)$.

ABSTRACT

Let $p$ be a prime and let $a$ be a positive integer. In this paper we investigate $\sum_{k=0}^{p^a-1}\binom[(h+1)k,k+d]/m^k$ modulo a prime $p$, where $d$ and $m$ are integers with $-h

Motivation & Objective

• To extend known congruences for sums of binomial coefficients and Catalan numbers modulo prime powers, particularly for $\sum \binom{3k}{k}/m^k$ and $\sum C_k^{(3)}/5^k$ modulo $p^a$.
• To derive exact evaluations of such sums using third- and fourth-order linear recurrence sequences and properties of cubic residues.
• To generalize previous results by Zhao, Pan, Sun, and Sun-Tauraso on sums involving $\binom{3k}{k}$, $C_k^{(2)}$, and $\binom{4k}{k}$ modulo $p$.
• To establish connections between Jacobi symbols, recurrence sequences, and the structure of sums involving generalized Catalan numbers of order 3.

Proposed method

• The paper employs third- and fourth-order linear recurrence sequences $u_n$ and $v_n$ defined by characteristic polynomials related to $(1+x)^3 - mx^2$ and $(1+x)^4 - (4^4/3^3)x^3$, respectively.
• It uses the theory of cubic residues and the cubic Jacobi symbol $\left(\frac{\cdot}{n}\right)_3$ to analyze the structure of sums modulo $p$.
• The method involves expressing sums of the form $\sum_{k=0}^{p^a-1} \binom{3k}{k+d}/m^k$ in terms of recurrence sequences $u_n$ and $v_n$, leveraging identities modulo $p$.
• Fermat's Little Theorem and properties of Lucas sequences are applied to reduce indices modulo $p^a$, particularly using $u_{n-p^a} \equiv \text{rational function of } u_n, u_{n+1} \pmod{p}$.
• The paper uses the discriminant of cubic and quartic polynomials to analyze irreducibility and apply Stickelberger’s theorem to determine Legendre symbol values.
• Key identities are derived using generating functions and symmetric polynomial identities, particularly relating $\sum \binom{4k}{k}/5^k$ and $\sum \bar{C}_k^{(3)}/5^k$ to linear combinations of recurrence terms.

Experimental results

Research questions

• RQ1 What is the value of $\sum_{k=0}^{p^a-1} \frac{\binom{3k}{k}}{5^k} \pmod{p}$ for odd prime $p$ and $a \in \mathbb{Z}^+$?
• RQ2 How do sums involving higher-order Catalan numbers $C_k^{(3)}$ and $\bar{C}_k^{(3)}$ behave modulo $p$ when weighted by $1/5^k$?
• RQ3 Can the sum $\sum_{k=0}^{p^a-1} \frac{\binom{4k}{k}}{5^k}$ be expressed in terms of recurrence sequences modulo $p$?
• RQ4 What is the role of the Legendre symbol $\left(\frac{-2}{p^a}\right)$ in determining the congruence class of $\sum \binom{4k}{k}/5^k$ modulo $p$?
• RQ5 Under what conditions does the sum $\sum_{k=1}^{p^a-1} \frac{\binom{3k}{k}}{6^k(k+1)} \equiv 0 \pmod{p}$ hold for $p^a \equiv 1 \pmod{6}$?

Key findings

• The sum $\sum_{k=0}^{p^a-1} \frac{C_k^{(3)}}{5^k} \equiv 1 \pmod{p}$, where $C_k^{(3)} = \frac{1}{4}\binom{4k}{k} - 3\binom{4k}{k-1}$, is established via recurrence reduction and Jacobi symbol analysis.
• The sum $\sum_{k=0}^{p^a-1} \frac{\bar{C}_k^{(3)}}{5^k} \equiv 3 \pmod{p}$, where $\bar{C}_k^{(3)} = 3\binom{4k}{k} - \binom{4k}{k+1}$, is derived using symmetric combinations of recurrence sequences.
• For $m=5$, the sum $\sum_{k=0}^{p^a-1} \frac{\binom{4k}{k}}{5^k} \equiv \frac{44 + \left(\frac{-2}{p^a}\right)}{288} \pmod{p}$, with the value depending on the Legendre symbol $\left(\frac{-2}{p^a}\right)$.
• The sum $\sum_{k=0}^{p^a-1} \frac{3^{3k}}{4^{4k}} \binom{4k}{k} \equiv \frac{44 + \left(\frac{-2}{p^a}\right)}{288} \pmod{p}$ is derived using a fourth-order recurrence and modular reduction of indices.
• The sum $\sum_{k=0}^{p^a-1} \frac{3^{3k}}{4^{4k}} \binom{4k}{k-1} \equiv -\frac{220 + 23\left(\frac{-2}{p^a}\right)}{864} \pmod{p}$ is obtained via linear combinations of recurrence terms.
• For $p>3$ and $p^a \equiv 1 \pmod{6}$, the sums $\sum_{k=1}^{p^a-1} \frac{\binom{3k}{k}}{6^k(k+1)} \equiv 0 \pmod{p}$ and $\sum_{k=1}^{p^a-1} \frac{\binom{3k}{k-1}}{6^k} \equiv 0 \pmod{p}$ are proven using irreducibility and Stickelberger’s theorem.