Discrete m-functions with Doubly Palindromic Continued Fraction Coefficients

TL;DR

This paper explores the special algebraic relationships of discrete m-functions with eventually periodic continued fractions, revealing that such relationships occur precisely when the continued fraction coefficients are doubly palindromic, characterized by specific symmetry properties.

Contribution

It establishes a novel connection between algebraic relations of m-functions and the doubly palindromic structure of continued fraction coefficients, a previously unrecognized symmetry condition.

Findings

Discrete m-functions with periodic continued fractions relate algebraically to their second solutions.

Doubly palindromic continued fractions are characterized by specific symmetry and concatenation properties.

The algebraic relationship holds if and only if the continued fraction coefficients are doubly palindromic.

Abstract

We demonstrate that discrete m-functions with eventually periodic continued fraction coefficients have an algebraic relationship to their second solution if and only if the periodic part of the sequence of continued fraction coefficients is doubly palindromic. In this setting, doubly palindromic means that each sequence is a repeated concatenation of two palindromes and a compatibility condition between the lengths of these palindromes is satisfied.