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+\delta_{ jk}]_{ 1\leq j,k\leq n}}&=1+\frac{ n^2(n^2-1)}{ 12}, \\ \det{ [|j-k|+\delta_{ 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 $b\not=0$ and $a^2\not=4b$, and the sequence $(w_m)_{ m\in\mathbb Z}$ with $w_{ k+1}=aw_k-bw_{ k-1}$ for all $k\in\mathbb Z$, we establish the identity $$\det[w_{ j-k}+c\delta_{ jk}]_{ 1\le j,k\le n} =c^n+c^{ n-1}nw_0+c^{ n-2}(w_1^2-aw_0w_1+bw_0^2)\frac{ u_n^2b^{ 1-n}-n^2}{ a^2-4b},$$ where $u_0=0$, $u_1=1$ and $u_{ k+1}=au_k-bu_{ k-1}$ for all $k=1,2,\ldots$.
Categories: math.NT