The Formal Inverse of the Period-Doubling Sequence

If $p$ is a prime number, consider a $p$-automatic sequence $(u_n)_{n\ge 0}$, and let $U(X) = \sum_{n\ge 0} u_n X^n \in \mathbb{F}_p[[X]]$ be its generating function. Assume that there exists a formal power series $V(X) = \sum_{n\ge 0} v_n X^n \in \mathbb{F}_p[[X]]$ which is the compositional inverse of $U$, i.e., $U(V(X))=X=V(U(X))$. The problem investigated in this paper is to study the properties of the sequence $(v_n)_{n\ge 0}$. The work was first initiated for the Thue-Morse sequence, and more recently the case of two variations of the Baum-Sweet sequence has been treated. In this paper, we deal with the case of the period-doubling sequence. We first show that the sequence of indices at which the period-doubling sequence takes value $0$ (resp., $1$) is not $k$-regular for any $k\ge 2$. Secondly, we give recurrence relations for its formal inverse, then we easily show that it is $2$-automatic, and we also provide an automaton that generates it. Thirdly, we study the sequence of indices at which this formal inverse takes value $1$, and we show that it is not $k$-regular for any $k\ge 2$ by connecting it to the characteristic sequence of Fibonacci numbers. We leave as an open problem the case of the sequence of indices at which this formal inverse takes value $0$. We end the paper with a remark on the case of generalized Thue-Morse sequences.