Hankel determinants of a Sturmian sequence

Haocong Song; Wen Wu; Haocong Song; Wen Wu

doi:10.3934/math.2022235

AIMS Mathematics

2022, Volume 7, Issue 3: 4233-4265. doi: 10.3934/math.2022235

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Research article

Hankel determinants of a Sturmian sequence

Haocong Song ,
Wen Wu ^,

School of Mathematics, South China University of Technology, Guangzhou 510640, China

Received: 05 September 2021 Revised: 09 December 2021 Accepted: 13 December 2021 Published: 20 December 2021
MSC : 11B75, 11C20

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 $.
- Hankel determinants,
- Sturmian sequences
Citation: Haocong Song, Wen Wu. Hankel determinants of a Sturmian sequence[J]. AIMS Mathematics, 2022, 7(3): 4233-4265. doi: 10.3934/math.2022235

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Abstract

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 $.

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