Evaluation of two determinants involving $q$-integers
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918509704904704 |
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| author | Sun, Zhi-Wei |
| author_facet | Sun, Zhi-Wei |
| contents | The $q$-analogue of an integer $m$ is given by $[m]_q=(1-q^m)/(1-q)$. Let $a$ be an integer, and let $n$ be a positive odd integer. Via discrete Fourier transforms, we establish the following two identities: $$\det\left[\left[\left\lfloor\frac{aj-(a+1)k}n\right\rfloor\right]_q\right]_{1\leqslant j,k\leqslant n}=-\left(\frac{a(a+1)}n\right)q^{(1-3n)/2}$$ and $$\det\left[\left[\left\lceil\frac{(a+1)j-ak}n\right\rceil\right]_q\right]_{1\leqslant j,k\leqslant n}=\left(\frac{a(a+1)}n\right)q^{(n-1)/2},$$ where $(\frac{\cdot}n)$ denotes the Jacobi symbol. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_16240 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Evaluation of two determinants involving $q$-integers Sun, Zhi-Wei Combinatorics Number Theory 05A30, 05A19, 11A15, 11C20, 15A15 The $q$-analogue of an integer $m$ is given by $[m]_q=(1-q^m)/(1-q)$. Let $a$ be an integer, and let $n$ be a positive odd integer. Via discrete Fourier transforms, we establish the following two identities: $$\det\left[\left[\left\lfloor\frac{aj-(a+1)k}n\right\rfloor\right]_q\right]_{1\leqslant j,k\leqslant n}=-\left(\frac{a(a+1)}n\right)q^{(1-3n)/2}$$ and $$\det\left[\left[\left\lceil\frac{(a+1)j-ak}n\right\rceil\right]_q\right]_{1\leqslant j,k\leqslant n}=\left(\frac{a(a+1)}n\right)q^{(n-1)/2},$$ where $(\frac{\cdot}n)$ denotes the Jacobi symbol. |
| title | Evaluation of two determinants involving $q$-integers |
| topic | Combinatorics Number Theory 05A30, 05A19, 11A15, 11C20, 15A15 |
| url | https://arxiv.org/abs/2605.16240 |