Linearly Unrelated Sequences

There are not many new results concerning the linear independence of numbers. Exceptions in the last decade are, e.g., the result of Sorokin [8] which proves the linear independence of logarithmus of special rational numbers, or that of Bezivin [2] which proves linear independence of roots of special functional equations. The algebraic independence of numbers can be considered as a generalization of linear independence. One can find many results of this nature. For instance, in [4] Bundschuh proves that if the special series of rational numbers converges to infinity very fast then they are algebraically independent. In [7] I prove a similar result for continued fractions. In that paper the so-called continued fractional algebraic independence of sequences was also defined. If we consider irrationality as a special case of linear independence then we can obtain many results. For instance, in [1] Apery proves the irrationality of ζ(3) and in [3] Borwein proves the irrationality of the sum ∑∞ n=1 1/(q n+ r), where q and r are integers such that q > 1 and r 6= 0. In 1975 Erdös defined the so-called irrationality of sequences in [5] (we will consider a generalization of this definition in Section 3) and in the same paper he proves the irrationality of the sequence {22n}. In 1993 in [6] I proved: