Euler, Pisot, Prouhet–Thue–Morse, Wallis and the duplication of sines

Abstract

We all know Euler’s product ${\prod(1+X^{2^n}) = (1-X)^{-1}}$ and its companion ${\prod(1-X^{2^n}) = \sum \pm X^j}$ , where the sequence of signs is the so-called Prouhet–Thue–Morse automatic sequence. Discussing generalizations of these two formulae, we are led respectively (1) to Wallis’ famous infinite product for π, (2) to a characterization of Pisot numbers, (3) to multigrade equalities and the Prouhet–Tarry–Escott problem, (4) to the product ${\prod_{0\leq j \leq n}{\rm sin}(2^j x)}$ and its sequence of signs as x runs through the intervals ${(j \pi/2^n, (j+1) \pi/2^n)}$, ${j \in [0, 2^n-1]}$ , (5) and finally to the Gelfond and Newman-Slater product and its generalization ${\prod \sin r^j x}$ , which plays a rôle in several papers when r = 2.