Confluent Vandermonde matrices, divided differences, and Lagrange-Hermite interpolation over quaternions

TL;DR

This paper introduces quaternion-valued confluent Vandermonde matrices, explores their connection to Lagrange-Hermite interpolation, and provides formulas for their rank and divided differences, extending classical interpolation concepts to quaternions.

Contribution

It presents the first study of confluent Vandermonde matrices with quaternion entries and develops new formulas for divided differences and rank in this context.

Findings

Derived the rank formula for quaternion confluent Vandermonde matrices

Established a representation formula for quaternion divided differences

Extended results to formal power series over quaternions

Abstract

We introduce the notion of a confluent Vandermonde matrix with quaternion entries and discuss its connection with Lagrange-Hermite interpolation over quaternions. Further results include the formula for the rank of a confluent Vandermonde matrix, the representation formula for divided differences of quaternion polynomials and their extensions to the formal power series setting.