Infinite products related to generalized Thue–Morse sequences.

Given an integer q ≥ 2 and θ 1 , ... , θ q - 1 ∈ { 0 , 1 } , let (θ n) n ≥ 0 be the generalized Thue–Morse sequence, defined to be the unique fixed point of the morphism 0 ↦ 0 θ 1 ⋯ θ q - 1 1 ↦ 1 θ ¯ 1 ⋯ θ ¯ q - 1 beginning with θ 0 : = 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) θ n and ∏ n = 1 ∞ (R (n) ) θ n. This generalizes relevant results given by Allouche, Riasat and Shallit in 2019 on infinite products related to the famous Thue–Morse sequence (t n) n ≥ 0 of the forms ∏ n = 1 ∞ (R (n) ) (- 1) t n and ∏ n = 1 ∞ (R (n) ) t n. [ABSTRACT FROM AUTHOR]