Formal inverses of the generalized Thue-Morse sequences and variations of the Rudin-Shapiro sequence

Łukasz Merta

doi:10.23638/DMTCS-22-1-15

Łukasz Merta - Formal inverses of the generalized Thue-Morse sequences and variations of the Rudin-Shapiro sequence

dmtcs:4954 - Discrete Mathematics & Theoretical Computer Science, May 25, 2020, vol. 22 no. 1 - https://doi.org/10.23638/DMTCS-22-1-15

Formal inverses of the generalized Thue-Morse sequences and variations of the Rudin-Shapiro sequenceArticle

Authors: Łukasz Merta

A formal inverse of a given automatic sequence (the sequence of coefficients of the composition inverse of its associated formal power series) is also automatic. The comparison of properties of the original sequence and its formal inverse is an interesting problem. Such an analysis has been done before for the Thue{Morse sequence. In this paper, we describe arithmetic properties of formal inverses of the generalized Thue-Morse sequences and formal inverses of two modifications of the Rudin{Shapiro sequence. In each case, we give the recurrence relations and the automaton, then we analyze the lengths of strings of consecutive identical letters as well as the frequencies of letters. We also compare the obtained results with the original sequences.

https://doi.org/10.23638/DMTCS-22-1-15

Source: arXiv.org:1810.03533

Volume: vol. 22 no. 1

Section: Automata, Logic and Semantics

Published on: May 25, 2020

Accepted on: April 8, 2020

Submitted on: November 5, 2018

Keywords: Mathematics - Number Theory,Mathematics - Combinatorics,11B83, 11B85

Licence: arXiv.org - Non-exclusive license to distribute