Finding both, the continued fraction and the Laurent series expansion of golden ratio analogs in the field of formal power series

TL;DR

This paper investigates formal power series analogs of the golden ratio, analyzing their continued fraction and Laurent series expansions through Hankel matrices derived from their coefficients.

Contribution

It introduces a method to study these analogs by examining Hankel matrices associated with their Laurent series coefficients, providing new insights into their structure.

Findings

Explicit continued fraction expansions derived

Laurent series expansions characterized

Hankel matrices reveal structural properties

Abstract

The focus of this paper is on formal power series analogs of the golden ratio. We are interested in both their continued fractions expansions as well as their Laurent series expansions. Our approach studies the Hankel matrices that are determined using the coefficients of the Laurent series expansions.