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

摘要

We all know Euler’s product Õ(1+X2n) = (1-X)-1{prod(1+X^{2^n}) = (1-X)^{-1}} and its companion Õ(1-X2n) = å±Xj{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 Õ_{0 £ j £ n}sin(2j x){prod_{0leq j leq n}{rm sin}(2^j x)} and its sequence of signs as x runs through the intervals (j p/2n, (j+1) p/2n){(j pi/2^n, (j+1) pi/2^n)}, j Î [0, 2n-1]{j in [0, 2^n-1]} , (5) and finally to the Gelfond and Newman-Slater product and its generalization Õsinrj x{prod sin r^j x} , which plays a rôle in several papers when r = 2.