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Powers of the Thue--Morse Series: 2-Adic Valuations and Automatic Odd Parts
Abstract
Let T(x)=∏j≥ 0(1−x2j)=∑n≥ 0t(n)xn be the Thue--Morse generating function, and write T(x)m=∑n≥ 0tm(n)xn for a positive integer m. For a nonzero integer a, let ν2(a) be the exponent of $2$ in a and put odd(a)=a/2ν2(a). For every s≥ 1, we prove that the sequence (odd(tm(n))bmod 2s)n≥ 0 is $2$-automatic when m=2r, r≥ 1, or m=3· 2r, r≥ 2. In both families, ν2(tm(n))=ν2binomn+m−1m−1. The valuation identity for m=2r is known; our proof recovers it and also establishes the automaticity assertion. For m=6 we prove ν2(t6(n))=ν2binomn+55+1n≡ 3pmod 4 and show that its odd parts modulo every 2s are $2$-automatic.
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BibTeX
@misc{shen2026powers,
title = {{Powers of the Thue--Morse Series: 2-Adic Valuations and Automatic Odd Parts}},
author = {Zhao Shen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16966},
eprint = {2609.16966},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16966},
}
RIS
TY - UNPB
TI - Powers of the Thue--Morse Series: 2-Adic Valuations and Automatic Odd Parts
AU - Zhao Shen
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16966
UR - https://arxiv.org/abs/2609.16966
AB - Let T(x)=∏_j≥ 0(1−x^2^j)=∑_n≥ 0t(n)x^n be the Thue--Morse generating function, and write T(x)^m=∑_n≥ 0t_m(n)x^n for a positive integer m. For a nonzero integer a, let ν_2(a) be the exponent of $2$ in a and put odd(a)=a/2^ν_2(a). For every s≥ 1, we prove that the sequence (odd(t_m(n))bmod 2^s)_n≥ 0 is $2$-automatic when m=2^r, r≥ 1, or m=3· 2^r, r≥ 2. In both families, ν_2(t_m(n))=ν_2binomn+m−1m−1. The valuation identity for m=2^r is known; our proof recovers it and also establishes the automaticity assertion. For m=6 we prove ν_2(t_6(n))=ν_2binomn+55+1_n≡ 3pmod 4 and show that its odd parts modulo every 2^s are $2$-automatic.
ER -