Vector Continued Fractions using a Generalised Inverse

TL;DR

This paper introduces a novel framework for vector continued fractions using a generalized inverse, establishing their relation to vector polynomials and demonstrating their strong convergence properties both analytically and numerically.

Contribution

It develops a new class of vector continued fractions based on a generalized inverse, extending classical scalar continued fractions to vector spaces with convergence guarantees.

Findings

Vector continued fractions can be constructed using a generalized inverse.

Vector Jacobi fractions relate to vector and scalar polynomial functions.

The convergence of these fractions is both analytically proven and numerically demonstrated.

Abstract

A real vector space combined with an inverse for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale. The choice of sign for different components of the vector inverse permits construction of vector analogues of the Jacobi continued fraction. These vector Jacobi fractions are related to vector and scalar-valued polynomial functions of the vectors, which satisfy recurrence relations similar to those of orthogonal polynomials. The vector Jacobi fraction has strong convergence properties which are demonstrated analytically, and illustrated numerically.