Irrationality and transcendence of continued fractions with algebraic integers

Abstract

We extend a result of Hančl, Kolouch and Nair on the irrationality and transcendence of continued fractions. We show that for a sequence $\{{\alpha }_n\}$ of algebraic integers of bounded degree, each attaining the maximum absolute value among their conjugates and satisfying certain growth conditions, the condition $$\underset{n\to \infty }{\mathrm{limsup}}|{\alpha }_n{|}^{\frac{1}{D{d}^{n-1}{\prod }_{i=1}^{n-2}(D{d}^i+1)}}=\infty$$ implies that the continued fraction $\alpha =[0;{\alpha }_1,{\alpha }_2,\dots ]$ is not an algebraic number of degree less than or equal to $D$.