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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.

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