On the transcendence of certain power series of algebraic numbers. (English) Zbl 0255.10038

Let \(a_k\) \((k=0,1,2,\ldots)\) be non-zero algebraic numbers. Put \[ A_k = \max_{i=0,\ldots,k} \overline{\vert a_i\vert}\quad (\text{house of }a_i) \quad\text{and }S_k = [\mathbb Q(a_0, a_1, \ldots, a_k) : \mathbb Q\}. \] Assume that \(M_k\) is a positive integer such that \(M_ka_i\) is an algebraic integer for \(i= 0,1,\ldots, k\). Let \(e\ge 0, e_1, e_2,\ldots\) be an increasing sequence of integers such that the gap series \(\sigma(z)= \sum_{k=0}^\infty a_kz^{e_k}\) has a positive radius of convergence \(R\). The main result in the present paper is the following: If \[ \lim (e_k + \log M_k + \log A_k)S_k/e_{k-1} = 0, \] then \(\sigma(\vartheta)\) is transcendental for every algebraic number \(\vartheta\) with \(0 < \vert\vartheta\vert<R\).
In the special case that the numbers \(a_k\) are rationals, this result was proved by H. Cohn [Bull. Am. Math. Soc. 52, 1042–1045 (1946; Zbl 0063.00941)]. For algebraic integer coefficients \(a_k\) the same assertion was proved by G. Baron and E. Braune [Compos. Math. 22, 1–6 (1970; Zbl 0212.39202)] under rather restrictive conditions.
Reviewer: P. L. Cijsouw


11J81 Transcendence (general theory)
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