## 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

### MSC:

 11J81 Transcendence (general theory)

### Citations:

Zbl 0063.00941; Zbl 0212.39202
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