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A formal inverse of a given automatic sequence (the sequence of coefficientsof the composition inverse of its associated formal power series) is alsoautomatic. The comparison of properties of the original sequence and its formalinverse is an interesting problem. Such an analysis has been done before forthe Thue{Morse sequence. In this paper, we describe arithmetic properties offormal inverses of the generalized Thue-Morse sequences and formal inverses oftwo modifications of the Rudin{Shapiro sequence. In each case, we give therecurrence relations and the automaton, then we analyze the lengths of stringsof consecutive identical letters as well as the frequencies of letters. We alsocompare the obtained results with the original sequences.

Formal inverses of the generalized Thue-Morse sequences and variations of the Rudin-Shapiro sequence