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Algebraic independence of the values of certain series by Mahler’s method. (English) Zbl 0764.11029

Suppose that \(f_ 1(z),\dots,f_ m(z)\) are algebraically independent functions of a complex variable satisfying \(f_ i(z)=a_ i(z)f_ i(Tz)+b_ i(z)\), where \(a_ i(z)\), \(b_ i(z)\) are rational functions and \(Tz=p(z^{-1})^{-1}\) for a polynomial \(p(z)\) of degree \(d\) larger than 1. Let \(\alpha\) be an algebraic number. We show that \(f_ 1(\alpha),\dots,f_ m(\alpha)\) are algebraically independent under suitable conditions on \(f\) and \(\alpha\). The transformation \(Tz\) is a generalization of the transformation \(Tz=z^ d\), which is somehow the “classical” transformation in Mahler’s method. As an application of our result, we deduce three corollaries, which extend earlier work by J. L. Davison and J. O. Shallit and J. Tamura.
The proof of the main result is based on Philippon’s criterion for algebraic indepencence.

MSC:

11J85 Algebraic independence; Gel’fond’s method
11J81 Transcendence (general theory)
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References:

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