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Linear independence of values of a certain generalisation of the exponential function – a new proof of a theorem of Carlson. (English) Zbl 1096.11024

The main result of this paper reads as follows. Let \(Q\in \mathbb{Z}[x]\), of degree \(q\geq 1\), satisfy \(Q(k)\neq 0\) for each integer \(k\geq 1\), and define the entire transcendental function \(G\) by \(G(x):=\sum_{n=0}^\infty x^n/\prod_{k=1}^n Q(k)\). Then, for any distinct \(\alpha_1,\dots ,\alpha_h\in \mathbb{Q}^\times\), the \(1+hq\) numbers \((\ast): 1,G(\alpha_1),\dots ,G^{(q-1)}(\alpha_1),\dots , G(\alpha_h),\dots ,G^{(q-1)}(\alpha_h)\) are linearly independent over \(\mathbb{Q}\). This assertion is a corollary of a slightly more general (but also purely qualitative) result of F. Carlson [Ark. Mat. Astron. Fys. A 25, No. 7, 1–13 (1935; Zbl 0011.39202)]. For the proof, the present author uses the Hilbert-Perron-Skolem method as explained, e.g., by himself and the reviewer [Abh. Math. Sem. Univ. Hamb. 69, 103–122 (1999; Zbl 0961.11022)]. This method requires very delicate divisibility considerations but only rather simple analytic tools. It should be noted that A. I. Galochkin [Mosc. Univ. Math. Bull. 34, 26–31 (1979); translation from Vestn. Mosk. Univ., Ser. I Mat. Mekh. 1979, No. 1, 26–30 (1979; Zbl 0413.10028)] proved very sharp linear independence measures for the numbers \((\ast)\) using a different and much more analytic method.

MSC:

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

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