Guided by Schwarz’ Functions: A Walk Through the Garden of Mahler’s Transcendence Method

Peter Bundschuh () and Keijo Väänänen ()
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Peter Bundschuh: Universität zu Köln, Mathematisches Institut
Keijo Väänänen: University of Oulu, Department of Mathematical Sciences

A chapter in From Arithmetic to Zeta-Functions, 2016, pp 91-101 from Springer

Abstract: 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.

Keywords: Algebraic independence of functions; Mahler’s method; Primary 11J91; Secondary 11B39 (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-28203-9_6

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DOI: 10.1007/978-3-319-28203-9_6

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