On determinants involving $(\frac{j^2-k^2}p)$ and $(\frac{jk}p)$

TL;DR

This paper investigates determinants involving Legendre symbols of quadratic expressions modulo an odd prime, revealing conditions under which these determinants are independent of certain parameters and exploring their properties.

Contribution

It introduces new results on determinants involving Legendre symbols, including conditions for independence from parameters and explicit evaluations for specific cases.

Findings

Determinant independence from parameter w when p ≡ 3 mod 4 and δ=0

Explicit evaluation of determinants for certain primes

New insights into Legendre symbol-based determinants

Abstract

Let $p$ be an odd prime and let $(\frac{\cdot}p)$ be the Legendre symbol. In this paper, we study the determinant $$\det\left[\left(\frac{j^2-k^2}p\right)+\left(\frac{jk}p\right)w\right]_{\delta\le j,k\le (p-1)/2}$$ with $\delta\in\{0,1\}$. For example, we prove that the determinant does not depend on $w$ if $p\equiv3\pmod4$ and $\delta=0$.