Transcendence of some power series for Liouville number arguments (Q723461)

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.