论文标题
评估一些toeplitz型决定因素
Evaluations of some Toeplitz-type determinants
论文作者
论文摘要
在本文中，我们评估了一些Toeplitz型决定因素。令$ n> 1 $为整数。我们证明了以下两个基本身份：\ begin {align*} \ det {[[J-k+δ_{Jk}] _ {1 \ leq j，k \ leq n}}＆= 1+ \ frac {n^2（n^2-1） \ det {[| j-k |+δ_{jk}] _ {1 \ leq j，k \ leq n}}}＆= \ begin {cases} \ frac {1+（ - 1）^{（n-1）^{（n-1）/2} n} n} n} n} \ frac {1+（-1）^{n/2}} {2}＆\ text {if} \ 2 \ 2 \ 2 \ mid n，\ end {cases} \ end {align {align*}其中$Δ__{jk} $是kronecker delta。对于复数$ a，b，c $带$ b \ not = 0 $和$ a^2 \ not = 4b $，以及序列$（w_m）_ {m \ in \ mathbb z} $，带有$ w_ {k+1} = aw_k-bw_ {k-bw_ {k-bw_ {k-bw_ {k-1} $，用于所有$ k-\ in \ kathbb z $，我们建立了\ mathbb z $，我们的身份$$ \ det [w_ {j-k}+cδ_{jk}] _ {1 \ le j，k \ le n} = c^n+c^{n-1} nw_0+c^{n-2}（w_1^2-aw_0w_1+bw_0^2） $ u_ {k+1} = au_k-bu_ {k-1} $ for ash last $ k = 1,2，\ ldots $。
In this paper we evaluate some Toeplitz-type determinants. Let $n>1$ be an integer. We prove the following two basic identities: \begin{align*} \det{[j-k+δ_{jk}]_{1\leq j,k\leq n}}&=1+\frac{n^2(n^2-1)}{12}, \\ \det{[|j-k|+δ_{jk}]_{1\leq j,k\leq n}}&= \begin{cases} \frac{1+(-1)^{(n-1)/2}n}{2}&\text{if}\ 2\nmid n,\\ \frac{1+(-1)^{n/2}}{2}&\text{if}\ 2\mid n, \end{cases} \end{align*} where $δ_{jk}$ is the Kronecker delta. For complex numbers $a,b,c$ with $b\not=0$ and $a^2\not=4b$, and the sequence $(w_m)_{m\in\mathbb Z}$ with $w_{k+1}=aw_k-bw_{k-1}$ for all $k\in\mathbb Z$, we establish the identity $$\det[w_{j-k}+cδ_{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 $k=1,2,\ldots$.
