Abstract
In this paper, transcendence results and, more generally, results on the algebraic independence of functions and their values are proved via Mahler’s analytic method. Here the key point is that the functions involved satisfy certain types of functional equations as G d (z d) = G d (z) − z∕(1 − z) in the case of \(G_{d}(z):=\sum _{h\geq 0}z^{d^{h} }/(1 - z^{d^{h} })\) for d ∈ { 2, 3, 4, …}. In 1967, these particular functions G d (z) were arithmetically studied by W. Schwarz using Thue–Siegel–Roth’s approximation method.
Dedicated to the Memory of Professor Wolfgang Schwarz
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Notes
- 1.
Here and in the sequel, \(\overline{\mathbb{Q}}\) denotes the field of all complex algebraic numbers.
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Bundschuh, P., Väänänen, K. (2016). Guided by Schwarz’ Functions: A Walk Through the Garden of Mahler’s Transcendence Method. In: Sander, J., Steuding, J., Steuding, R. (eds) From Arithmetic to Zeta-Functions. Springer, Cham. https://doi.org/10.1007/978-3-319-28203-9_6
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DOI: https://doi.org/10.1007/978-3-319-28203-9_6
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