Evaluation of two determinants involving $q$-integers

Abstract

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: $$\det {\left[{\left[\lfloor \frac{aj-(a+1)k}{n}\rfloor \right]}_q\right]}_{1⩽j,k⩽n}=-\left(\frac{a(a+1)}{n}\right)q^{(1-3n)/2}$$ and $$\det {\left[{\left[\lceil \frac{(a+1)j-ak}{n}\rceil \right]}_q\right]}_{1⩽j,k⩽n}=\left(\frac{a(a+1)}{n}\right)q^{(n-1)/2},$$ where $(\frac{\cdot }{n})$ denotes the Jacobi symbol.

arXiv comment10 pages. Correct typos and make Theorem 1.3 more general