# More infinite products: Thue–Morse and the gamma function

Research paper by J.-P. Allouche, S. Riasat; J. Shallit

Indexed on: 07 Mar '18Published on: 23 Feb '18Published in: The Ramanujan Journal

#### Abstract

Letting $$(t_n)$$ denote the Thue–Morse sequence with values 0, 1, we note that the Woods–Robbins product \begin{aligned} \prod _{n \ge 0}\left( \frac{2n+1}{2n+2}\right) ^{(-1)^{t_n}} = 2^{-1/2} \end{aligned} involves a rational function in n and the ± 1 Thue–Morse sequence $$((-1)^{t_n})_{n \ge 0}$$ . The purpose of this paper is twofold. On the one hand, we try to find other rational functions for which similar infinite products involving the ± 1 Thue–Morse sequence have an expression in terms of known constants. On the other hand, we also try to find (possibly different) rational functions R for which the infinite product $$\prod R(n)^{t_n}$$ also has an expression in terms of known constants.