[Paper Review] Supercongruences and hypergeometric transformations

Chen Wang|arXiv (Cornell University)|Mar 22, 2020

Advanced Mathematical Identities2 citations

TL;DR

This paper proves two conjectural supercongruences of Sun using a ${}_4F_3$ hypergeometric transformation identity that relates sums of products of squared central binomial coefficients to rational hypergeometric series. For primes $p > 3$, it establishes congruences modulo $p^4$ linking these sums to Euler numbers $E_{p-3}$ and the Legendre symbol $(-1)^{(p-1)/2}$, confirming deep arithmetic properties of hypergeometric-type sums.

ABSTRACT

In this paper, we mainly prove two conjectural supercongruences of Sun by using the following identity $$ \sum_{k=0}^n\binom{2k}{k}^2\binom{2n-2k}{n-k}^2=16^n\sum_{k=0}^n\frac{\binom{n+k}{k}\binom{n}{k}\binom{2k}{k}^2}{(-16)^k} $$ which arises from a ${}_4F_3$ hypergeometric transformation. For any prime $p>3$, we prove that \begin{gather*} \sum_{n=0}^{p-1}\frac{n+1}{8^n}\sum_{k=0}^n\binom{2k}{k}^2\binom{2n-2k}{n-k}^2\equiv(-1)^{(p-1)/2}p+5p^3E_{p-3}\pmod{p^4}, \sum_{n=0}^{p-1}\frac{2n+1}{(-16)^n}\sum_{k=0}^n\binom{2k}{k}^2\binom{2n-2k}{n-k}^2\equiv(-1)^{(p-1)/2}p+3p^3E_{p-3}\pmod{p^4}, \end{gather*} where $E_{p-3}$ is the $(p-3)$th Euler number.

Motivation & Objective

• To prove two conjectural supercongruences posed by Sun involving sums of products of squared central binomial coefficients.
• To establish arithmetic congruences modulo $p^4$ for these sums when $p > 3$ is prime.
• To connect the resulting congruences to Euler numbers $E_{p-3}$ and the Legendre symbol $(-1)^{(p-1)/2}$.

Proposed method

• Derive a key identity relating $\sum_{k=0}^n\binom{2k}{k}^2\binom{2n-2k}{n-k}^2$ to a rational hypergeometric series via a ${}_4F_3$ transformation.
• Use the derived identity to express the sum in terms of $\sum_{k=0}^n\frac{\binom{n+k}{k}\binom{n}{k}\binom{2k}{k}^2}{(-16)^k}$.
• Apply $p$-adic analysis and known properties of binomial coefficients modulo $p^4$ to evaluate the sums over $n = 0$ to $p-1$.
• Leverage known congruences for harmonic sums and Bernoulli numbers to relate the results to Euler numbers $E_{p-3}$.
• Use the Legendre symbol $(-1)^{(p-1)/2}$ to capture the sign dependence on $p \mod 4$ in the final congruences.
• Verify the congruences modulo $p^4$ through algebraic manipulation and known $p$-adic identities for hypergeometric series.

Experimental results

Research questions

• RQ1 How can the hypergeometric transformation identity be used to prove supercongruences involving sums of squared central binomial coefficients?
• RQ2 What is the $p$-adic behavior of the sum $\sum_{n=0}^{p-1}\frac{n+1}{8^n}\sum_{k=0}^n\binom{2k}{k}^2\binom{2n-2k}{n-k}^2$ modulo $p^4$?
• RQ3 How do Euler numbers $E_{p-3}$ emerge in the congruence structure of these hypergeometric sums for prime $p > 3$?
• RQ4 Can the sign in the congruence be expressed via $(-1)^{(p-1)/2}$, reflecting the quadratic character of $p$?
• RQ5 What is the precise form of the supercongruence for the alternating version of the sum with denominator $(-16)^n$?

Key findings

• For any prime $p > 3$, the sum $\sum_{n=0}^{p-1}\frac{n+1}{8^n}\sum_{k=0}^n\binom{2k}{k}^2\binom{2n-2k}{n-k}^2$ is congruent to $(-1)^{(p-1)/2}p + 5p^3E_{p-3}$ modulo $p^4$.
• The sum $\sum_{n=0}^{p-1}\frac{2n+1}{(-16)^n}\sum_{k=0}^n\binom{2k}{k}^2\binom{2n-2k}{n-k}^2$ satisfies the congruence $(-1)^{(p-1)/2}p + 3p^3E_{p-3}$ modulo $p^4$.
• The Euler number $E_{p-3}$ appears explicitly in both supercongruences, linking the sums to arithmetic properties of Bernoulli and Euler numbers.
• The sign of the linear term in $p$ is governed by $(-1)^{(p-1)/2}$, reflecting the quadratic residue status of $p$ modulo 4.
• The results are established using a non-trivial ${}_4F_3$ hypergeometric transformation identity that connects two different representations of the same sum.
• The congruences hold modulo $p^4$, indicating a deep arithmetic structure beyond modulo $p$ or $p^2$.