Seminar-workshop in mathematics,yaoundé, cameroon,December 10–15,2001

This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of “mathematical community” is the broadest. We include “schools” of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.     

First announcement Fifth International Symposium on Generalized Convexity Luminy-Marseille,France, June 17–21, 1996

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First announcement Fifth International Symposium on Generalized Convexity Luminy-Marseille, France, June 17–21, 1996     

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

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.     

APPLIED MATHEMATICS–A JOURNAL OF CHINESE UNIVERSITIES Volume 32, Series B, 2017

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