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arXiv · 1709.03398

More Infinite Products: Thue-Morse and the Gamma function

Jean-Paul Allouche·Samin Riasat·Jeffrey Shallit
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Abstract

Letting $(t_n)$ denote the Thue-Morse sequence with values $0, 1$, we note that the Woods-Robbins product $$ \prod_{n \geq 0} \left(\frac{2n+1}{2n+2}\right)^{(-1)^{t_n}} = 2^{-1/2} $$ involves a rational function in $n$ and the $\pm 1$ Thue-Morse sequence $((-1)^{t_n})_{n \geq 0}$. 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 $\pm 1$ 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.

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Jean-Paul Allouche, Samin Riasat, Jeffrey Shallit. 2017-09-11. More Infinite Products: Thue-Morse and the Gamma function. https://arxiv.org/abs/1709.03398

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