Coquet-type formulas for the rarefied weighted Thue–Morse sequence

Newman proved for the classical Thue–Morse sequence, ((−1)^s(n))_n≥0, that for all N∈N with real constants satisfying c₂>c₁>0 and λ=log3/log4. Coquet improved this result and deduced , where F(x) is a nowhere-differentiable, continuous function with period 1 and η(N)∈{−1,0,1}. In this paper we obtain for the weighted version of the Thue–Morse sequence that for the sum a Coquet-type formula exists for every r∈{0,1,2} if and only if the sequence of weights is eventually periodic. From the specific Coquet-type formulas we derive parts of the weak Newman-type results that were recently obtained by Larcher and Zellinger.