Determinants of Binomial-Related Circulant Matrices

TL;DR

This paper introduces Binomial-related circulant matrices, explores their algebraic properties, and derives explicit determinant formulas for specific cases, expanding understanding of their structure and applications.

Contribution

It defines Binomial-related matrices, provides explicit determinant formulas for special cases, and discusses open problems in the algebraic study of circulant matrices.

Findings

Determinant formulas for z in {1, -1, i, -i}

Extension of known binomial circulant matrix results

Discussion of open problems in the field

Abstract

Due to their rich algebraic structures and various applications, circulant matrices have been of interest and continuously studied. In this paper, the notions of Binomial-related matrices have been introduced. Such matrices are circulant matrices whose the first row is the coefficients of $(x+zy)^n$, where $z$ is a complex number of norm $1$ and $n$ is a positive integer. In the case where $z\in \{1,-1,i,-i\}$, the explicit formula for the determinant of such matrices are completely determined. Known results on the determinants of binomial circulant matrices can be viewed as special cases. Finally, some open problems are discussed.