CRITERIA FOR APWENIAN SEQUENCES
Résumé
In 1998, Allouche, Peyrière, Wen and Wen showed that the Hankel determinant Hn of the Thue-Morse sequence over {−1, 1} satisfies Hn/2 n−1 ≡ 1 (mod 2) for all n ≥ 1. Inspired by this result, Fu and Han introduced apwenian sequences over {−1, 1}, namely, ±1 sequences whose Hankel determinants satisfy Hn/2 n−1 ≡ 1 (mod 2) for all n ≥ 1, and proved with computer assistance that a few sequences are apwenian. In this paper, we obtain an easy to check criterion for apwenian sequences, which allows us to determine all apwenian sequences that are fixed points of substitutions of constant length. Let f (z) be the generating functions of such apwenian sequences. We show that for all integer b ≥ 2 with f (1/b) = 0, the real number f (1/b) is transcendental and its irrationality exponent is equal to 2. Besides, we also derive a criterion for 0-1 apwenian sequences whose Hankel determinants satisfy Hn ≡ 1 (mod 2) for all n ≥ 1. We find that the only 0-1 apwenian sequence, among all fixed points of substitutions of constant length, is the period-doubling sequence. Various examples of apwenian sequences given by substitutions with projection are also provided. Furthermore, we prove that all Sturmian sequences over {−1, 1} or {0, 1} are not apwenian. And we conjecture that fixed points of substitution of non-constant length over {−1, 1} or {0, 1} can not be apwenian.
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