Generalized Palindromic Continued Fractions

TL;DR

This paper generalizes the concept of palindromic continued fractions to m-palindromes, providing a transcendence criterion, construction methods, and exploring their relationship with stammering continued fractions.

Contribution

It introduces m-palindromes as a new generalization, extends transcendence criteria, and offers construction techniques and connections to stammering continued fractions.

Findings

Established a transcendence criterion for m-palindromes.

Developed methods to construct examples of m-palindromes.

Explored the relationship between m-palindromes and stammering continued fractions.

Abstract

In this paper we introduce a generalization of palindromic continued fractions as studied by Adamczewski and Bugeaud. We refer to these generalized palindromes as $m$-palindromes, where $m$ ranges over the positive integers. We provide a simple transcendence criterion for $m$-palindromes, extending and slightly refining an analogous result of Adamczewski and Bugeaud. We also provide methods for constructing examples of $m$-palindromes. Such examples allow us to illustrate our transcendence criterion and to explore the relationship between $m$-palindromes and stammering continued fractions, another concept introduced by Adamczewski and Bugeaud.