2014/06/06 by Hao Fu, Fu, Hao, Guo-Niu Han +1
Computer Science · Mathematics · #05A05 #05A10 #05A15 #05A19 #11B50 #11B65 #11B85 #15A15 #Advanced Algebra and Geometry #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1406.1589
openalex publication_date 2014/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 integral numbers. We speak of t-extension when the entries along the diagonal in the Hankel determinant are all multiplied by~t. Then 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 \it only one to be an odd integral number. 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.