A note on the transcendental continued fractions (Q1092091)

Consider the continued fractions denoted by \(A=a_ 1+\frac{1}{a_ 2+}\frac{1}{a_ 3+}...\) and \(B=b_ 1+\frac{1}{b_ 2+}\frac{1}{b_ 3+}...\) with \(a_ 1\geq 0\), \(a_ n>0\) for all \(n>1\); \(b_ 1\geq 0\), \(b_ n>0\) for all \(n>1\). G. Nettler proved A, B, \(A\pm B\), A/B, A B are all transcendental numbers provided the relation \(a_ n>b_ n>a_{n- 1}^{[(n-1)^ 2]}\) holds for n sufficiently large. Here the author has got an interesting generalization in proving that the transcendence of these five numbers holds provided the relation \(a_ n>b_ n>a_{n-1}^{\gamma (n-1)}\) holds for n large enough, with \(\gamma\) a constant strictly superior to 16. His method consists of improving G. Nettler's method. As an example he shows his Theorem applies to A made with the sequence \(a_ n=2^{(2n)!}\) and B made with the sequence \(b_ n=2^{9\cdot [(2n-1)!]}.\)