On determinants involving (\fracj+kp)±(\fracj-kp)

2024/09/12 by Deyi Chen, Zhi‐Wei Sun, Chen, Deyi +1
Mathematics · #11A15 #11C20 #15A15 #15A18 #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Graph theory and applications #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2409.08213

openalex publication_date 2024/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p=2n+1 be an odd prime. In this paper, we mainly evaluate determinants involving (\frac j+kp)±(\fracj-kp), where (\frac⋅p) denotes the Legendre symbol. When p≡1\pmod4, we determine the characteristic polynomials of the matrices [(\fracj+kp)+(\fracj-kp)]_{1≤ j,k≤ n} and [(\fracj+kp)-(\fracj-kp)]_{1≤ j,k≤ n}, and also establish the general identity amp; |x+(\fracj+kp)+(\fracj-kp)+(\frac jp)y+(\frac kp)z+(\fracjkp)w|_{1≤ j,k≤ n}
=amp; (-p)^(p-5)/4((\fracp-12)²wx-(\fracp-12y-1)(\fracp-12z-1)).

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