Expansions of generalized Thue-Morse numbers

We introduce generalized Thue-Morse numbers of the form pi(beta)(theta) := Sigma(infinity)(n=1)theta(n)/beta(n) where beta is an element of (1, m+1] with m is an element of N and theta=(theta(n))(n >= 1) is an element of {0, 1, ... , m}(N) is a generalized Thue-Morse sequence previously studied by many authors in different terms. This is a natural generalization of the classical Thue-Morse number Sigma(infinity)(n=1) t(n)/2(n) where (t(n))(n >= 0) is the well-known Thue-Morse sequence 01101001 ... . We study when theta would be the unique, greedy, lazy, quasi-greedy and quasi-lazy beta-expansions of pi(beta)(theta), and generalize a result given by Kong and Li in 2015. In particular we deduce that the shifted Thue-Morse sequence (t(n))(n >= 1) is the unique beta-expansion of Sigma(infinity)(n=1)t(n)/beta(n) if and only if it is the greedy expansion, if and only if it is the lazy expansion, if and only if it is the quasi-greedy expansion, if and only if it is the quasi-lazy expansion, and if and only if beta is no less than the Komornik-Loreti constant. (c) 2022 Elsevier Inc. All rights reserved.