Mahler Functions and Transcendence

Transcendence of a fast converging series of rational numbers

“…In section 2, we will present a brief review of the history of fast converging series and their irrationality properties. As in this paper we are not interested in their transcendence properties, we will only mention Mahler's method [20] when it works. In section 3, we will present our new results, including irrationality statements and computation of irrationality measures.…”

Liouville sequences

“…This paper also contains a list of references including [6] and [8] which present the criteria for algebraic independence of certain Liouville series. A survey of these types of results can be found in the book of Nishioka [7]. Also the result of Petruska [9] establishes several interesting criteria concerning the strong Liouville numbers.…”

On the Transcendence of Some Infinite Series

“…(of Theorem 2.1) Let N be a sufficiently large positive integer satisfying (4) and (5). Assume that δ is a sufficiently small positive real number.…”

“…In [4] it is shown that if lim n→∞ log 3 log 2 a n > 1 and a n ∈ ‫ގ‬ for all n ∈ ‫ގ‬ then S is a transcendental number. Many other criteria for S to be transcendental can be found in [1], [5], [6] or [8] but divisibility properties or fullfilling special equations are necessary. It seems to be the case that in general it is not easy to decide when S is a transcendental number if lim sup n→∞ log 3 log 2 a n < 1 holds and divisibility properties or fullfilling certain equations are not required.…”