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

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.