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Euler, Pisot, Prouhet-Thue-Morse, Wallis and the duplication of sines. (English) Zbl 1169.11005

The authors first show the following simple identity \[ \prod_{0\leq k <n} (1+aX^{2^k})=\sum_{0\leq j <2^n} a^{s(j)} X^j, \] where \(a\) is a complex number and \(s(j)\) is the sum of binary digits of a nonnegative integer \(n\). Then they generalize it to obtain a similar formula for the product \(\prod_{0\leq k <n} (1+aX^{\lambda_k}),\) where \(\lambda_k,\) \(k=1,2,3,\dots,\) is a sequence of complex numbers. From these identities they derive several known results, like Wallis’ formula for \(\pi\), etc. They also show that the generalization of such formulas leads to other (unrelated) results concerning the Prouhet-Tarry-Escott problem, Pisot numbers, Gelfond’s bound on \[ \sup_{x \in \mathbb{R}}|\sin (x) \sin (2x) \dots \sin (2^n x)|, \] etc.

MSC:

11A63 Radix representation; digital problems
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11L99 Exponential sums and character sums
26D05 Inequalities for trigonometric functions and polynomials
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11B85 Automata sequences
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References:

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