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) = $\sum_{n\ge 0} u_nX^n ∈ F_p[[X]]$ be its generating function. Assume that there exists a formal power series $V(X) = \sum_{n\ge 0} v_n X^n ∈ F_p[[X]]$ 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 other sequences (variations of the Baum-Sweet sequence, variations of the Rudin-Shapiro sequence and generalized Thue-Morse sequences) 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 the value 0 (resp., 1) is not k-regular for any $k \ge 2$. Secondly, we give recurrence relations for its formal inverse, then we 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 the 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 the value 0.