2016/01/19 by Gawro, Maciej, Ulas, Maciej
#11B83 #11B85 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1601.04840
Let p be a prime number and consider a p-automatic sequence \bf u=(un)n∈\N and its generating function U(X)=∑n=0∞unXn∈\mathbbFp[[X]]. Moreover, let us suppose that u0=0 and u1≠ 0 and consider the formal power series V∈\mathbbFp[[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)\pmod2 and \bf t=(tn)n∈\N is the Prouhet-Thue-Morse sequence defined on the two letter alphabet \0,1\. More precisely, we study the sequence \bf c=(cn)n∈\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 cn=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 cn=0 is not k-regular for any k. Moreover, we present a result concerning some density properties of a sequence related to \bf a.