Evaluations of some Toeplitz-type determinants

Abstract

In this paper we evaluate some Toeplitz-type determinants. Let $n>1$ be an integer. We prove the following two basic identities: \begin{align*} \det{[j-k+δ_{jk}]_{1\leq j,k\leq n}}&=1+\frac{n^2(n^2-1)}{12}, \\ \det{[|j-k|+δ_{jk}]_{1\leq j,k\leq n}}&= \begin{cases} \frac{1+(-1)^{(n-1)/2}n}{2}&\text{if}\ 2\nmid n,\\ \frac{1+(-1)^{n/2}}{2}&\text{if}\ 2\mid n, \end{cases} \end{align*} where ${\delta }_{jk}$ is the Kronecker delta. For complex numbers $a,b,c$ with $b\ne 0$ and $a^2\ne 4b$, and the sequence $(w_m{)}_{m\in \mathbb{Z}}$ with $w_{k+1}=a{w}_k-b{w}_{k-1}$ for all $k\in \mathbb{Z}$, we establish the identity $$\det [w_{j-k}+c{\delta }_{jk}{]}_{1\le j,k\le n}=c^n+c^{n-1}n{w}_0+c^{n-2}(w_1^2-a{w}_0{w}_1+b{w}_0^2)\frac{u_n^2{b}^{1-n}-n^2}{a^2-4b},$$ where $u_0=0$, $u_1=1$ and $u_{k+1}=a{u}_k-b{u}_{k-1}$ for all $k=1,2,\dots$.

22 pages.Add parts (ii) and (iii) of Theorem 1.1