Infinite products related to generalized Thue-Morse sequences

作者：Li, Yao-Qiang^*

来源：Monatshefte für Mathematik, 2021, 194(3): 577-600.

DOI：10.1007/s00605-020-01480-x

• Thue-Morse sequence
• Morphic sequence
• Closed formulas for infinite products
• Woods and Robbins product

Given an integer q >= 2 and theta(1), ... , theta(q-1) is an element of{0,1}, let (theta(n))(n >= 0) be the generalized Thue-Morse sequence, defined to be the unique fixed point of the morphism @@@ 0 -> 0 theta(1) ... theta(q-1) @@@ 1 -> 1 (theta) over bar (1) ... (theta) over bar (q-1) @@@ beginning with theta(0) := 0, where (0) over bar := 1 and (1) over bar := 0. For ad hoc rational functions R, we evaluate infinite products of the forms @@@ Pi(infinity)(n=1)(R(n)((-1)theta n) and Pi(infinity)(n=1)(R(n)(theta 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 @@@ Pi(infinity)(n=1)(R(n))((-1)tn) and Pi(infinity)(n=1)(R(n))(tn).