Hankel determinant calculus for the Thue-Morse and related sequences

The Hankel determinants of certain automatic sequences f are evaluated modulo a prime number. In most cases, the Hankel determinants of automatic sequences do not have any closed-form expressions; the traditional methods, such as LU-decomposition and Jacobi continued fraction, cannot be applied directly. Our method is based on a simple idea: the Hankel determinants of each sequence g equal to f modulo p are equal to the Hankel determinants of f modulo p. The clue then consists of finding a nice sequence g, whose Hankel determinants have closed-form expressions. Several examples are presented, including a result saying that the Hankel determinants of the Thue-Morse sequence are nonzero, first proved by Allouche, Peyrière, Wen and Wen using determinant manipulation.