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On \(q\)-analogues of divergent and exponential series. (English) Zbl 1169.11031
Let \(\mathbb K\) be a number field. Assume that \(m\in \mathbb Z^+\). Suppose that \(a, q, \alpha_1, \dots , \alpha_m \in \mathbb K \setminus \{ 0\}\). Denote by \(f(t)\) each of the functions \(\sum_{n=0}^\infty a^n \prod_{j=1}^n (1-t^j)\), \(\sum_{n=0}^\infty t^n \prod_{j=1}^n (1-q^j)^{-1}\) and \(\prod_{n=0}^\infty (1-tq^n)\). Under the special conditions the author proves that the numbers \(1\), \(f(\alpha_1)\), …, \(f(\alpha_m)\) are linearly independent over \(\mathbb K\) and gives the bound for the measure. The proof makes use of Padé approximation.

MSC:
11J82 Measures of irrationality and of transcendence
11J72 Irrationality; linear independence over a field
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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