Generalized Pascal Triangles and Toeplitz Matrices

TL;DR

This paper investigates determinants of generalized Pascal matrices, expressing them via Toeplitz matrices, and applies this to evaluate determinants and generate Fibonacci and Lucas subsequences.

Contribution

It introduces a factorization of generalized Pascal matrices into Toeplitz matrices and unipotent triangular matrices, enabling determinant evaluation and subsequence generation.

Findings

Determinants of generalized Pascal matrices equal those of associated Toeplitz matrices.

Established formulas for determinants related to Fibonacci and Lucas subsequences.

Provided methods to generate linear Fibonacci and Lucas subsequences from matrix minors.

Abstract

The purpose of this article is to study determinants of matrices which are known as generalized Pascal triangles (see [1]). We present a factorization by expressing such a matrix as a product of a unipotent lower triangular matrix, a Toeplitz matrix and a unipotent upper triangular matrix. The determinant of a generalized Pascal matrix equals thus the determinant of a Toeplitz matrix. This equality allows us to evaluate a few determinants of generalized Pascal matrices associated to certain sequences. In particular, we obtain families of quasi-Pascal matrices whose principal minors generate any arbitrary linear subsequences F(nr+s) or L(nr+s), (n=1, 2, 3, ...) of Fibonacci or Lucas sequence.