2014/06/06 by Guo-Niu Han, Han, Guo-Niu
Computer Science · Mathematics · #05A10 #05A15 #11B50 #11B65 #11B85 #11C20 #11J82 #11Y65 #15A15 #30B70 #Advanced Combinatorial Mathematics #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1406.1586
openalex publication_date 2014/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Hankel determinants of certain automatic sequences f are evaluated, based on a calculation 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-decompo\-si\-tion 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. The present approach shortens the proof of the latter result significantly. We also prove that the corresponding Hankel determinants do not vanish when the powers 2n in the infinite product defining the ± 1 Thue--Morse sequence are replaced by 3n.