Construction of some special subsequences within a Farey sequence

TL;DR

This paper introduces a new mapping and algorithm to generate special subsequences within Farey sequences, revealing their structure and continued fraction properties, which are relevant for quantum soliton state analysis.

Contribution

It presents a novel mapping between subsequences of different Farey sequence orders and an algorithm to generate these subsequences, along with their continued fraction expansions.

Findings

Established a new connection between Farey subsequences of different orders

Developed an algorithm to generate all special subsequences within a Farey sequence

Derived continued fraction expansions for elements of these subsequences

Abstract

Recently it has been found that some special subsequences within a Farey sequence play a crucial role in determining the ranges of coupling constant for which quantum soliton states can exist for an integrable derivative nonlinear Schrodinger model. In this article, we find a novel mapping which connects two such subsequences belonging to Farey sequences of different orders. By using this mapping, we construct an algorithm to generate all of these special subsequences within a Farey sequence. We also derive the continued fraction expansions for all the elements belonging to a subsequence and observe a close connection amongst the corresponding expansion coefficients.