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Linear independence measures for infinite products. (English) Zbl 0999.11036
Let $$f$$ be an entire transcendental function with rational coefficients in its power series about the origin, satisfying a functional equation $$f(qz)=(z-c)f(z)+Q(z)$$ with $$q\in\mathbb Z$$, $$|q|\geq 2$$, $$c=\pm q^l$$, $$l\in\mathbb Z$$ and $$Q(z)\in\mathbb Q[z]$$. Let $$\alpha\in\mathbb Q^\times$$ with $$\alpha\neq\pm c q^k$$ for $$k=1,2,\dots$$. Then the numbers $$1$$, $$f(\alpha)$$ and $$f(-\alpha)$$ are linear independent over $$\mathbb Q$$. Moreover, a linear independence measure of these numbers is given in this paper. If $$c=-1$$ and $$Q(z)\equiv 0$$, then the function $$f$$ can be written as an infinite product, which has been extensively studied by several authors.

##### MSC:
 11J72 Irrationality; linear independence over a field 11J82 Measures of irrationality and of transcendence
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