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

Deyi Chen; Zhi-Wei Sun

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arXiv · 2408.14401

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

Deyi Chen·Zhi-Wei Sun

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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 $\delta=0$.

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Deyi Chen, Zhi-Wei Sun. 2024-08-26. On determinants involving $(\frac{j^2-k^2}p)$ and $(\frac{jk}p)$. https://arxiv.org/abs/2408.14401

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