Transcendence measure of values of functions satisfying certain functional equations

Arithmetical properties of certain infinite products

“…Assume that there exists a positive integer c ≥ 2 such that for all n ∈ N we have E N (z)/F N (z) = 1 with some N satisfying n ≤ N < n + c, and that for any E k (z)/F k (z) = 1 with k large enough the zeros and poles of E k (z)/F k (z) are real, if r ≥ 3, and positive, if r = 2. Then, for any algebraic number α with 0 < |α| < 1 satisfying (3), Φ(α) is a transcendental number having the measure (5).…”

“…Let α with 0 < |α| < 1 be an algebraic number satisfying (3). The following argument for the proof of Theorem 4, which involves several lemmas, can be viewed as a variant of the argument used in Galochkin [5] (see also [1]). Let us denote…”

Transcendence measures by Mahler's transcendence method

“…of Lemma 4 in [3]. The hypothesis n 5 C{m) required there can be specified to n 2 m , if we restrict Galochkin's proof to the functions considered here.…”

“…For each non-negative integer k , we define Liouville estimates (see [3], Lemma 5) applied to ik) P, (C,a) and a (a) lead to the inequality of the lemma.…”

An improvement of a transcendence measure of Galochkin and Mahler's S-numbers

“…Let / ( z ) be a function which is transcendental over C(z) and holomorphic in some neighborhood U of the origin, and satisfies the functional equation In the notation as above, Mahler proved in [4] that the number / ( a ) is transcendental. In [2], Galochkin considered a quantitative version of this result and gave the following transcendence measure of f(a): THEOREM (Galochkin [2] Our main purpose is to sharpen this estimate. To state our results, we recall usual notation and the definition of Mahler's S-numbers (cf.…”