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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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