On determinants involving $(\frac{j+k}p)\pm(\frac{j-k}p)$
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913769751314432 |
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| author | Chen, Deyi Sun, Zhi-Wei |
| author_facet | Chen, Deyi Sun, Zhi-Wei |
| contents | 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*} |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_08213 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On determinants involving $(\frac{j+k}p)\pm(\frac{j-k}p)$ Chen, Deyi Sun, Zhi-Wei Number Theory 11A15, 11C20, 15A15, 15A18 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*} |
| title | On determinants involving $(\frac{j+k}p)\pm(\frac{j-k}p)$ |
| topic | Number Theory 11A15, 11C20, 15A15, 15A18 |
| url | https://arxiv.org/abs/2409.08213 |