Criteria for apwenian sequences

Abstract

In 1998, Allouche, Peyrière, Wen and Wen showed that the Hankel determinant Hn of the Thue-Morse sequence on {−1, 1} satisfies Hn/2 n−1 ≡ 1 (mod 2) for all n ≥ 1. Inspired by this phenomenon, Fu and Han introduced the apwenian sequence on {−1, 1}, namely, the ±1 sequences whose Hankel determinants satisfy Hn/2 n−1 ≡ 1 (mod 2) for all n ≥ 1, and proved that a few sequences are apwenian with computer assistance. 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 the irrationality exponent is equal to 2.Besides, we also derive a criterion for zero-one apwenian sequences whose Hankel determinants satisfy Hn ≡ 1 (mod 2) for all n ≥ 1. We find that the only zero-one 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 given. Furthermore, we prove that all Sturmian sequences on {−1, 1} or {0, 1} are not apwenian. And we conjecture that fixed points of substitution of non-constant length on {−1, 1} or {0, 1} can not be apwenian.