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

TL;DR

This paper investigates the arithmetic properties of formal inverses of generalized Thue-Morse sequences and modifications of the Rudin-Shapiro sequence, analyzing their automata, recurrence relations, and letter frequency characteristics.

Contribution

It provides a detailed analysis of the formal inverses of these sequences, including automaton construction, recurrence relations, and comparison with original sequences, which was not previously explored.

Findings

Automata and recurrence relations for formal inverses are derived.

Lengths of consecutive identical letters are analyzed.

Letter frequency distributions are compared with original sequences.

Abstract

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.