Recurrence Relations and Determinants

TL;DR

This paper explores the properties of a special class of determinants called n-determinants, revealing their connections to recurrence relations, Fibonacci numbers, and Schur functions, with implications for matrix minors and sequence analysis.

Contribution

It introduces n-determinants and establishes their relationships with recurrence sequences, Fibonacci numbers, and Schur functions, extending known determinant formulas.

Findings

1-determinants are upper Hessenberg determinants

Several 1-determinants equal Fibonacci numbers

Schur functions can be expressed as n-determinants

Abstract

We examine relationships between two minors of order n of some matrices of n rows and n+r columns. This is done through a class of determinants, here called $n$-determinants, the investigation of which is our objective. We prove that 1-determinants are the upper Hessenberg determinants. In particular, we state several 1-determinants each of which equals a Fibonacci number. We also derive relationships among terms of sequences defined by the same recurrence equation independently of the initial conditions. A result generalizing the formula for the product of two determinants is obtained. Finally, we prove that the Schur functions may be expressed as $n$-determinants.