On determinants involving $(\frac{j+k}p)\pm(\frac{j-k}p)$
Deyi Chen, Zhi-Wei Sun
TL;DR
This work analyzes determinants built from Legendre symbols applied to linear forms in the indices, focusing on matrices with entries $(\frac{j+k}{p})\pm(\frac{j-k}{p})$ for primes $p$. By deriving the eigenstructure of the resulting matrices $A_+$ and $A_-$ and employing Dirichlet characters, it determines complete characteristic polynomials and determinants in the cases $p\equiv1\pmod4$ and $p>3$, $p\equiv3\pmod4$, including connections to the class number $h(-p)$. The authors then extend these determinants to a general family with additional linear perturbations $(\frac{j}{p})y$, $(\frac{k}{p})z$, and $(\frac{jk}{p})w$, obtaining explicit formulas that depend on $p\bmod4$ and arithmetic constants such as $c_p$ and $d_p$, and involving the Dirichlet-character framework. An auxiliary determinant theorem is proved to handle these perturbations, and the results are used to prove further identities (Theorems 1.2 and 1.3), including a congruence for $d_p$ modulo 4 related to Sun S24 conjectures. Overall, the paper advances exact determinant evaluations for Legendre-symbol matrices and links them with algebraic number-theoretic invariants.
Abstract
Let $p=2n+1$ be an odd prime. In this paper, we mainly evaluate determinants involving $(\frac {j+k}p)\pm(\frac{j-k}p)$, where $(\frac{\cdot}p)$ denotes the Legendre symbol. When $p\equiv1\pmod4$, we determine the characteristic polynomials of the matrices $$\left[\left(\frac{j+k}p\right)+\left(\frac{j-k}p\right)\right]_{1\le j,k\le n}\ \ \text{and}\ \ \left[\left(\frac{j+k}p\right)-\left(\frac{j-k}p\right)\right]_{1\le j,k\le n},$$ and also establish the general identity \begin{align*} &\ \left|x+\left(\frac{j+k}p\right)+\left(\frac{j-k}p\right)+\left(\frac jp\right)y+\left(\frac kp\right)z+\left(\frac{jk}p\right)w\right|_{1\le j,k\le n} \\=&\ (-p)^{(p-5)/4}\left(\left(\frac{p-1}2\right)^2wx-\left(\frac{p-1}2y-1\right)\left(\frac{p-1}2z-1\right)\right). \end{align*}