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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., Comment: 20 pages

Studying and comparing arithmetic properties of a given automatic sequence and the sequence of coefficients of the composition inverse of the associated formal power series (the formal inverse of that sequence) is an interesting problem. This problem was studied before for the Thue-Morse sequence. In this paper, we study arithmetic properties of the formal inverses of two sequences closely related to the well-known Baum-Sweet sequence. We give the recurrence relations for their formal inverses and we determine whether the sequences of indices at which these formal inverses take value $0$ and $1$ are regular. We also show an unexpected connection between one of the obtained sequences and the formal inverse of the Thue-Morse sequence., Comment: 25 pages

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 ${\bf a}$., Comment: 16 pages; revised version will appear in Discrete Mathematics

In a recent preprint on ArXiv, Bacher introduced a twisted version of the Stern sequence. His paper contains in particular three conjectures relating the generating series for the Stern sequence and for the twisted Stern sequence. Soon afterwards Coons published two papers in {\it Integers}: first he proved these conjectures, second he used his result to obtain a correlation-type identity for the Stern sequence. We recall here a simple result of Reznick and we state a similar result for the twisted Stern sequence. We deduce an easy proof of Coons' identity, and a simple proof of Bacher's conjectures. Furthermore we prove identities similar to Coons' for variations on the Stern sequence that include Bacher's sequence.

In this paper we construct certain type of near-optimal approximations of a class of analytic functions in the unit disc by power series with two distinct coefficients. More precisely, we show that if all the coefficients of the power series f(z) are real and lie in [-a,a] where a < 1, then there exists a power series Q(z) with coefficients in {-1,+1} such that |f(z)-Q(z)| approaches 0 at the rate exp(-C/|1-z|) as z approaches 1 non-tangentially inside the unit disc. A result by Borwein-Erdelyi-Kos shows that this type of decay rate is best possible. The special case f=0 yields a near-optimal solution to the "fair duel" problem of Konyagin., Comment: 8 pages, 1 figure