Evaluations of some Toeplitz-type determinants

TL;DR

This paper evaluates specific Toeplitz-type determinants, providing explicit formulas and identities for various cases, including those involving Kronecker delta and recurrence sequences, advancing the understanding of their structure.

Contribution

The paper introduces new explicit formulas for Toeplitz-type determinants, including identities involving Kronecker delta and recurrence relations, which were not previously documented.

Findings

Derived explicit formulas for determinants with Kronecker delta modifications.

Established identities for determinants involving recurrence sequences.

Provided formulas applicable to complex parameters and recurrence relations.

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+\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 $k=1,2,\ldots$.