On determinants involving $(\frac{j+k}p)\pm(\frac{j-k}p)$

TL;DR

This paper evaluates determinants involving Legendre symbols for matrices related to an odd prime p, determining characteristic polynomials and establishing a general determinant identity for specific matrix forms.

Contribution

It provides explicit characteristic polynomials and a new general determinant identity for matrices constructed from Legendre symbols when p ≡ 1 mod 4.

Findings

Determined characteristic polynomials for specific matrices involving Legendre symbols.

Established a general determinant identity involving parameters x,y,z,w for matrices of size n.

Derived explicit formulas for determinants when p ≡ 1 mod 4.

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*}