On formal inverse of the Prouhet-Thue-Morse sequence

Let p be a prime number and consider a p-automatic sequence u=(un)n∈N and its generating function U(X)=∑n=0∞unXn∈Fp[[X]]. Moreover, let us suppose that u0=0 and u1≠0 and consider the formal power series V∈Fp[[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 un=tn, where tn=s2(n)(mod2) and t=(tn)n∈N is the Prouhet-Thue-Morse sequence defined on the two letter alphabet {0,1}. More precisely, we study the sequence c=(cn)n∈N which is the sequence of coefficients of the compositional inverse of the generating function of the sequence t. This sequence is clearly 2-automatic. We describe the sequence a characterizing solutions of the equation cn=1. In particular, we prove that the sequence a is 2-regular. We also prove that an increasing sequence characterizing solutions of the equation cn=0 is not k-regular for any k. Moreover, we present a result concerning some density properties of a sequence related to a.