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More infinite products: Thue–Morse and the gamma function

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Abstract

Letting \((t_n)\) denote the Thue–Morse sequence with values 0, 1, we note that the Woods–Robbins product

$$\begin{aligned} \prod _{n \ge 0}\left( \frac{2n+1}{2n+2}\right) ^{(-1)^{t_n}} = 2^{-1/2} \end{aligned}$$

involves a rational function in n and the ± 1 Thue–Morse sequence \(((-1)^{t_n})_{n \ge 0}\). 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 ± 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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Acknowledgements

This paper is an extended version of [11]. While we were preparing this extended version, we found the paper [15] which has interesting results on finite (and infinite) sums involving the sum of digits of integers in integer bases.

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Allouche, JP., Riasat, S. & Shallit, J. More infinite products: Thue–Morse and the gamma function. Ramanujan J 49, 115–128 (2019). https://doi.org/10.1007/s11139-017-9981-7

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