Transcendence of some power series for Liouville number arguments. (English) Zbl 1437.11100
Let \(\{ a_n\}_{n=1}^\infty\) and \(\{ b_n\}_{n=1}^\infty\) be sequences of rational integers such that \(a_n>1\) for all positive integers \(n\). Assume that
\(\limsup_{n\to\infty}\frac{\log\mid b_n\mid}{\log a_n} <1<
\liminf_{n\to\infty}\frac{\log a_{n+1}}{\log a_n}\). Let \(\alpha\) be a Liouville number. Under the special conditions the author proves that the sum of the series \(\sum_{n=1}^\infty\frac{b_n}{a_n}\alpha^n\) eighter rational number or transcendental number.
The author obtains similar results when \(b_n\) belong to finite algebraic number field and for \(p\)-adic cases.
The author obtains similar results when \(b_n\) belong to finite algebraic number field and for \(p\)-adic cases.
Reviewer: Jaroslav Hančl (Ostrava)
MSC:
| 11J81 | Transcendence (general theory) |
| 11J17 | Approximation by numbers from a fixed field |
| 11J61 | Approximation in non-Archimedean valuations |
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