Open AccessHankel determinants of a Sturmian sequence
Author(s) -
Haocong Song,
AUTHOR_ID,
Wu Wen
Publication year - 2022
Publication title -
aims mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.329
H-Index - 15
ISSN - 2473-6988
DOI - 10.3934/math.2022235
Subject(s) - mathematics , combinatorics , alphabet , partition (number theory) , hankel matrix , sequence (biology) , integer (computer science) , discrete mathematics , mathematical analysis , chemistry , computer science , philosophy , linguistics , biochemistry , programming language
Let $ \tau $ be the substitution $ 1\to 101 $ and $ 0\to 1 $ on the alphabet $ \{0, 1\} $. The fixed point of $ \tau $ obtained starting from 1, denoted by $ {\bf{s}} $, is a Sturmian sequence. We first give a characterization of $ {\bf{s}} $ using $ f $-representation. Then we show that the distribution of zeros in the determinants induces a partition of integer lattices in the first quadrant. Combining those properties, we give the explicit values of the Hankel determinants $ H_{m, n} $ of $ {\bf{s}} $ for all $ m\ge 0 $ and $ n\ge 1 $.