Infinite products related to generalized Thue–Morse sequences

Overview & Synopsis

Given an integer q≥2 and θ1,…,θq−1∈{0,1}, let (θn)n≥0 be the generalized Thue–Morse sequence, defined to be the unique fixed point of the morphism 0↦1↦0θ1⋯θq−11θ¯¯¯1⋯θ¯¯¯q−1 beginning with θ0:=0 , where 0¯¯¯:=1 and 1¯¯¯:=0 . For ad hoc rational functions R, we evaluate infinite products of the forms ∏n=1∞(R(n))(−1)θnand∏n=1∞(R(n))θn. This generalizes relevant results given by Allouche, Riasat and Shallit in 2019 on infinite products related to the famous Thue–Morse sequence (tn)n≥0 of the forms ∏n=1∞(R(n))(−1)tnand∏n=1∞(R(n))tn.