On t-extensions of the Hankel determinants of certain automatic sequences

Abstract. In 1998, Allouche, Peyrière, Wen and Wen considered the Thue–Morse sequence, and proved that all the Hankel determinants of the period-doubling sequence are odd integers. We speak of t-extension when the entries along the diagonal in the Hankel determinant are all multiplied by t. We prove that the t-extension of each Hankel determinant of the period-doubling sequence is a polynomial in t, whose leading coefficient is the only one to be an odd integer. Our proof makes use of the combinatorial set-up developed by Bugeaud and Han, which appears to be very suitable for this study, as the parameter t counts the number of fixed points of a permutation. Finally, we prove that all the t-extensions of the Hankel determinants of the regular paperfolding sequence are polynomials in t of degree less than or equal to 3.