On formal inverse of the Prouhet-Thue-Morse sequence

TL;DR

This paper investigates the arithmetic properties of the inverse of the Prouhet-Thue-Morse sequence's generating function, revealing its 2-regularity and contrasting it with the non-regularity of related zero-solution sequences.

Contribution

It introduces the study of the inverse sequence's coefficients, characterizes solutions to specific equations, and establishes regularity and density properties of these sequences.

Findings

The inverse sequence's solution set for c_n=1 is 2-regular.

The sequence for solutions of c_n=0 is not k-regular for any k.

Density properties of related sequences are analyzed.

Abstract

Let $p$ be a prime number and consider a $p$-automatic sequence ${\bf u}=(u_{n})_{n\in\N}$ and its generating function $U(X)=\sum_{n=0}^{\infty}u_{n}X^{n}\in\mathbb{F}_{p}[[X]]$. Moreover, let us suppose that $u_{0}=0$ and $u_{1}\neq 0$ and consider the formal power series $V\in\mathbb{F}_{p}[[X]]$ which is a compositional inverse of $U(X)$, i.e., $U(V(X))=V(U(X))=X$. In this note we initiate the study of arithmetic properties of the sequence of coefficients of the power series $V(X)$. We are mainly interested in the case when $u_{n}=t_{n}$, where $t_{n}=s_{2}(n)\pmod{2}$ and ${\bf t}=(t_{n})_{n\in\N}$ is the Prouhet-Thue-Morse sequence defined on the two letter alphabet $\{0,1\}$. More precisely, we study the sequence ${\bf c}=(c_{n})_{n\in\N}$ which is the sequence of coefficients of the compositional inverse of the generating function of the sequence ${\bf t}$. This sequence is clearly 2-automatic. We describe the sequence ${\bf a}$ characterizing solutions of the equation $c_{n}=1$. In particular, we prove that the sequence ${\bf a}$ is 2-regular. We also prove that an increasing sequence characterizing solutions of the equation $c_{n}=0$ is not $k$-regular for any $k$. Moreover, we present a result concerning some density properties of a sequence related to ${\bf a}$.